Multicriteria decision-making method using the Dice similarity measure based on the reduct intuitionistic fuzzy sets of interval-valued intuitionistic fuzzy sets

This paper proposes the concept of the reduct intuitionistic fuzzy sets of interval-valued intuitionistic fuzzy sets (IVIFSs) with respect to adjustable weight vectors and the Dice similarity measure based on the reduct intuitionistic fuzzy sets to explore the effects of optimism, neutralism, and pessimism in decision making. Then a decision-making method with the pessimistic, optimistic, and neutral schemes desired by the decision maker is established by combining adjustable weight vectors and the Dice similarity measure for IVIFSs. The proposed decision-making method is more flexible and adjustable in practical problems and can determine the ranking order of alternatives and the optimal one(s), so that it can overcome the difficulty of the ranking order and decision making when there exist the same measure values of some alternatives in some cases. This adjustable feature can provide the decision maker with more selecting schemes and actionable results for the decision-making analysis. Finally, two illustrative examples are employed to show the feasibility of the proposed method in practical applications.     

A method of inference in approximate reasoning based on interval-valued fuzzy sets

This paper introduces and discusses a method of approximate inference which operates on the extension of the concept of a fuzzy set by the concept of an interval-valued fuzzy set. This method allows a formal, fuzzy representation to be built for verbal decision algorithms. Furthermore, it can have an effective computer representation. An example showing how this method operates is provided.     

区间值模糊推理的三Ⅰ算法

模糊推理在控制和人工智能等领域已得到了广泛的应用，但其理论基础还不完善，王国俊教授提出的模糊命题逻辑的形式演绎系统和三Ⅰ算法为模糊推理奠定了严格的逻辑基础，本文把三Ⅰ算法用于区间值模糊推理，并且指出一般模糊推理是在区间退化为点时的特殊区间值模糊推理，从而把一般模糊推理纳入于区间值模糊推理的框架之内。     

The similarity measure of generalized fuzzy numbers based on interval distance

In this paper, we proposed a new interval distance of two fuzzy numbers that satisfy on metric properties. Also, this metric distance satisfies on some of the other properties. Then, we used this metric for similarity measure. Finality, we tested with some examples.     

A multi-attribute group decision-making method based on weighted geometric aggregation operators of interval-valued trapezoidal fuzzy numbers

With respect to the multiple attribute group decision making problems in which the attribute values take the form of generalized interval-valued trapezoidal fuzzy numbers (GITFN), this paper proposed a decision making method based on weighted geometric aggregation operators. First, some operational rules, the distance and comparison between two GITFNs are introduced. Second, the generalized interval-valued trapezoidal fuzzy numbers weighted geometric aggregation (GITFNWGA) operator, the generalized interval-valued trapezoidal fuzzy numbers ordered weighted geometric aggregation (GITFNOWGA) operator, and the generalized interval-valued trapezoidal fuzzy numbers hybrid geometric aggregation (GITFNHGA) operator are proposed, and their various properties are investigated. At the same time, the group decision methods based on these operators are also presented. Finally, an illustrate example is given to show the decision-making steps and the effectiveness of this method.     

A novel approach to interval-valued intuitionistic fuzzy soft set based decision making

The soft set theory, originally proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. The interval-valued intuitionistic fuzzy soft set is a combination of an interval-valued intuitionistic fuzzy set and a soft set. The aim of this paper is to investigate the decision making based on interval-valued intuitionistic fuzzy soft sets. By means of level soft sets, we develop an adjustable approach to interval-valued intuitionistic fuzzy soft sets based decision making and some numerical examples are provided to illustrate the developed approach. Furthermore, we also define the concept of the weighted interval-valued intuitionistic fuzzy soft set and apply it to decision making.     

A group decision making approach based on aggregating interval data into interval-valued intuitionistic fuzzy information

Group decision making is one of the most important problems in decision making sciences. The aim of this article is to aggregate the interval data into the interval-valued intuitionistic fuzzy information for multiple attribute group decision making. In this model, the decision information is provided by decision maker, which is characterized by interval data. Based on the idea of mean and variance in statistics, we first define the concepts of satisfactory and dissatisfactory intervals of attribute vector against each alternative. Using these concepts, we develop an approach to aggregate the attribute vector into interval-valued intuitionistic fuzzy number under group decision making environment. A practical example is provided to illustrate the proposed method. To show the validity of the reported method, comparisons with other methods are also made.     

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.     

Fractional programming approach to fuzzy weighted average

This paper proposes a fractional programming approach to construct the membership function for fuzzy weighted average. Based on the -cut representation of fuzzy sets and the extension principle, a pair of fractional programs is formulated to find the -cut of fuzzy weighted average. Owing to the special structure of the fractional programs, in most cases, the optimal solution can be found analytically. Consequently, the exact form of the membership function can be derived by taking the inverse function of the -cut. For other cases, a discrete but exact solution to fuzzy weighted average is provided via an efficient solution method. Examples are given for illustration.     

Perturbation of fuzzy sets and fuzzy reasoning based on normalized Minkowski distances

In this paper, we extend the concept of the perturbation of fuzzy sets based on normalized Minkowski distances and present some new conclusions on perturbation raised by various operations of fuzzy sets. These operations are induced by triangular norms and conorms. Furthermore, we discuss the perturbation of fuzzy reasoning.     

Formalization of implication based fuzzy reasoning method

Fuzzy reasoning includes a number of important inference methods for addressing uncertainty. This line of fuzzy reasoning forms a common logical foundation in various fields, such as fuzzy logic control and artificial intelligence. The full implication triple I method (a method only based on implication, TI method for short) for fuzzy reasoning is proposed in 1999 to improve the popular CRI method (a hybrid method based on implication and composition). The current paper delves further into the TI method, and a sound logical foundation is set for the TI method based on the monoidal t-norm based logical system MTL.     

An improved method on group decision making based on interval-valued intuitionistic fuzzy prioritized operators

Interval-valued intuitionistic fuzzy prioritized operators are widely used in group decision making under uncertain environment due to its flexibility to model uncertain information. However, there is a shortcoming in the existing aggregation operators (interval-valued intuitionistic fuzzy prioritized weighted average (IVIFPWA)) to deal with group decision making in some extreme situations. For example, when an expert gives an absolute negative evaluation, the operators could lead to irrational results, so that they are not effectively enough to handle group decision making. In this paper, several examples are illustrated to show the unreasonable results in some of these situations. Actually, these unreasonable cases are common for operators in dealing with product averaging, not only emerging in IVIFPWA operators. To overcome the shortcoming of these kinds of operators, an improvement of making slight adjustment on initial evaluations is provided. Numerical examples are used to show the efficiency of the improvement.     

Note on continuous interval-valued intuitionistic fuzzy aggregation operator

Recently, Zhou et al. proposed the C-IVIFOWA operator for aggregating interval-valued intuitionistic fuzzy numbers. The flaws of the C-IVIFOWA operator are illustrated by two counter-examples. The aim of this short communication is to show how the existing aggregation operator can be improved to increase accuracy. Some desirable properties of the revised C-IVIFOWA operator are studied in detail.     

Operations and measurement of interval-valued fuzzy numbers based on the method of structured element

In this paper we introduced the definition of interval-valued fuzzy numbers based on the definition of interval-valued fuzzy set and the operations of real fuzzy numbers by extension principle, defined the operations of interval-valued fuzzy numbers and their distance, and gave their structured element representations and operations of them by using the fuzzy structured element method. The work of this paper is an expansion of the research of real fuzzy numbers.