Zhen-Song Chen

Proportional hesitant fuzzy linguistic term set for multiple criteria group decision making

We propose the general concept of PHFLTS.We define the negation, union, and intersection operations on PHFLTSs.We present the PHFLWA and PHFLOWA operators.A transformation algorithm is proposed to convert the proportional comparative linguistic pairs into PHFLTSs.We develop a proportional hesitant fuzzy linguistic MCGDM model. The theory of hesitant fuzzy linguistic term sets (HFLTSs) is a powerful technique used to describe hesitant situations, which are typically assessed by experts using several possible linguistic values or rich expressions instead of a single term. The union of HFLTSs with respect to each expert, that is, an extended HFLTS (EHFLTS), further facilitates the elicitation of linguistic assessments for addressing group decision-making problems because EHFLTSs can deal with generalized (either consecutive or non-consecutive) linguistic terms. In this study, we propose proportional HFLTSs (PHFLTSs), which include the proportional information of each generalized linguistic term. The mathematical form for a PHFLTS is consistent with that for a linguistic distribution assessment. However, the underlying meanings of the proportions associated with generalized linguistic terms are different. PHFLTSs can be viewed as a special method for performing linguistic distribution assessments. PHFLTSs are recognized as a useful extension of HFLTSs and a possibility distribution for HFLTSs under different assumptions. We define the basic operations with closed properties among PHFLTSs on the basis of t-norms and t-conorms. We then propose a probability theory-based outranking method for PHFLTSs by providing possibility degree formulas. We also study two fundamental aggregation operators for PHFLTSs, namely, the proportional hesitant fuzzy linguistic weighted averaging operator and the proportional hesitant fuzzy linguistic ordered weighted averaging operator. Several important properties of these aggregation operators are investigated. Finally, we use the proposed multiple criteria group decision-making model in practical applications.

A Note on Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets

In this note, we point out an error to the proof of Theorem 3.4 in Zhang and Xu (Int J Intell Syst 2014;29(12):1061–1078) by a counterexample. We find that the inequality (i.e., |(πβ1)2−(πβ2)2|≤|(πβ1)2−(πβ3)2| ) with respect to the degrees of indeterminacy of any three Pythagorean fuzzy numbers in the proof of Theorem 3.4 in Zhang and Xus paper is not valid. A new proof is provided in this note.

Generating HFLTS possibility distribution with an embedded assessing attitude

Possibility distribution provides an alternative explanation of linguistic terms in an hesitant fuzzy linguistic term set (HFLTS). It is generated from a given HFLTS based on the assumption that all linguistic terms in an HFLTS follow a uniform distribution. In decision making contexts, linguistic terms in each HFLTS are assumed to exhibit the same possibility to represent the decision-maker(DM)s assessment rating. However, DMs may give ratings based on different assessing attitudes because of individual differences in cognitive styles. In practice, comparative linguistic expressions are generally converted into HFLTSs through the transformation process as DMs usually use comparative linguistic expressions instead of HFLTSs when giving ratings. The transformed HFLTSs likewise exclude the DMs assessing attitudes. To enhance the interpretability of generated possibility distributions, this paper relaxes the original assumption of uniform distribution and incorporates an attitudinal dimension into the transformation process that converts comparative linguistic expressions to HFLTSs. With these modified conditions for possibility distribution generation, an assessing attitude-driven approach is proposed based on probability density functions (PDFs) to generate HFLTS possibility distributions. The proposed PDF-based HFLTS possibility distributions personalize individual semantics and further facilitate the process of computing with words to obtain assessing attitude-embedded accurate linguistic results that are easy for individuals to interpret and understand.

On Extending Power-Geometric Operators to Interval-Valued Hesitant Fuzzy Sets and Their Applications to Group Decision Making

Developing aggregation operators for interval-valued hesitant fuzzy sets (IVHFSs) is a technological task we are faced with, because they are specifically important in many problems related to the fusion of interval-valued hesitant fuzzy information. This paper develops several novel kinds of power geometric operators, which are referred to as variable power geometric operators, and extends them to interval-valued hesitant fuzzy environments. A series of generalized interval-valued hesitant fuzzy power geometric (GIVHFG) operators are also proposed to aggregate the IVHFSs to model mandatory requirements. One of the important characteristics of these operators is that objective weights of input arguments are variable with the change of a non-negative parameter. By adjusting the exact value of the parameter, the influence caused by some “false” or “biased” arguments can be reduced. We demonstrate some desirable and useful properties of the proposed aggregation operators and utilize them to develop techniques for multiple criteria group decision making with IVHFSs considering the heterogeneous opinions among individual decision makers. Furthermore, we propose an entropy weights-based fitting approach for objectively obtaining the appropriate value of the parameter. Numerical examples are provided to illustrate the effectiveness of the proposed techniques.

Triangular intuitionistic fuzzy random decision making based on combination of parametric estimation, score functions, and prospect theory

This study investigates and improves the operational laws of triangular intuitionistic fuzzy numbers. The triangular intuitionistic fuzzy random variable (TIFRV) is introduced on the basis of the concepts of the triangular intuitionistic fuzzy number and triangular fuzzy random variable. Related properties of a TIFRV are also proposed and verified. To solve the problem of multi-criteria decision making on aspiration levels—a situation in which criterion weights are unknown and criterion values are given in terms of TIFRVs—this study proposes a triangular intuitionistic fuzzy random decision- making method based on a combination of parametric estimation, score functions, and prospect theory. In this method, the decision maker evaluates alternatives with triangular intuitionistic fuzzy numbers in different periods of decision making and thus enables the estimation of the parameters of the triangular intuitionistic fuzzy population and the creation of an intuitionistic triangular fuzzy random matrix. An expectation-variance intuitionistic fuzzy matrix is constructed on the basis of mean-variance analysis, and a fuzzy random score function is then defined to transform a normalized expectation-variance intuitionistic fuzzy matrix into a score function matrix. Prospect theory is used to calculate the values of prospect score functions, and the information entropy method is used to determine criterion weights. This procedure generates comprehensive prospect score function values that determine the final ranking of alternatives. A practical example is presented to show the feasibility and effectiveness of the proposed approach.

On Generalized Extended Bonferroni Means for Decision Making

The extended Bonferroni mean (EBM) recently proposed differs from the classical Bonferroni mean, as it aims to capture the heterogeneous interrelationship among the attributes instead of presupposing a homogeneous relation among them. In this study, we generalize the EBM to explicitly and profoundly understand its aggregation mechanism by defining a composite aggregation function. We adopt the approach of optimizing the choice of weighting vectors for the generalized EBM (GEBM) with respect to the least absolute deviation of residuals. We also investigate several desirable properties of the GEBM. Our special interest in this study is to investigate the ability of the GEBM to model mandatory requirements. Finally, the influence of replacing the conjunctive of the GEBM is analyzed to show how the change of the conjunctive affects the global andness and orness of the GEBM. Meanwhile, the aggregation mechanism of the EBM is specified and provided with quite intuitive interpretations for application.

A Framework for Triangular Fuzzy Random Multiple-Criteria Decision Making

Most real-world decisions practically occur in extremely complex environments characterized by both fuzziness and randomness. This phenomenon highlights the requirement for new evaluation methods in a fuzzy random environment and new ways to address fuzzy random multiple-criteria decision-making (MCDM) problems. This study reviews fuzzy random variable (FRV) to evaluate fuzzy random decision-making environment. Given the inaccuracy of certain precision formulas proposed in previous studies for the variance of a triangular FRV, this work presents the detailed process of calculating precision variance formulas and discusses several properties of the expectation and variance of triangular FRVs (TFRVs). The united variance of a TFRV vector is also proven to possess non-additive properties. Thus, an ordered weighted averaging (OWA) operator is extended to aggregate fuzzy random data by proposing a triangular fuzzy random OWA operator. Motivated by the idea of mean–variance analysis, an expectation-variance-based method is employed to rank TFRVs. Furthermore, a novel triangular fuzzy random MCDM method is developed, and certain numerical examples are provided to demonstrate the ability of TFRVs to comprehensively assess the performance of a specific alternative. This work also illustrates how the triangular fuzzy random MCDM framework can be extended to any fuzzy random decision-making process.

A Novel MAGDM Approach With Proportional Hesitant Fuzzy Sets

In this paper, we propose an extension of hesitant fuzzy sets, i.e., proportional hesitant fuzzy sets (PHFSs), with the purpose of accommodating proportional hesitant fuzzy environments. The components of PHFSs, which are referred to as proportional hesitant fuzzy elements (PHFEs), contain two aspects of information provided by a decision-making team: the possible membership degrees in the hesitant fuzzy elements and their associated proportions. Based on the PHFSs, we provide a novel approach to addressing fuzzy multi-attribute group decision making (MAGDM) problems. Different from the traditional approach, this paper first converts fuzzy MAGDM (expressed by classical fuzzy numbers) into proportional hesitant fuzzy multi-attribute decision making (represented by PHFEs), and then solves the latter through the proposal of a proportional hesitant fuzzy TOPSIS approach. In this process, preferences of the decision-making team are calculated as the proportions of the associated membership degrees. Finally, a numerical example and a comparison are provided to illustrate the reliability and effectiveness of the proposed approach.

Commentary on A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems

In this paper, we demonstrate that the Theorem 3.1 in a recently published paper by Garg [Appl. Soft Comput. 38 (2016) 988999] is incorrect by a counterexample. Further, we point out the shortcomings of Gargs proposed generalized improved score function for practical decision making.

Interval-valued Pythagorean Fuzzy Frank Power Aggregation Operators based on An Isomorphic Frank Dual Triple

Interval-valued Pythagorean fuzzy sets (PFSs), as an extension of PFSs, have strong potential in the management of complex uncertainty in real-world applications. This study aims to develop several intervalvalued Pythagorean fuzzy Frank power (IVPFFP) aggregation operators with an adjustable parameter via the integration of an isomorphic Frank dual triple. First, a special automorphism on unit interval is introduced to construct an isomorphic Frank dual triple; and this triple is further applied on the definition of interval-valued Pythagorean fuzzy Frank operational laws. Second, two IVPFFP aggregation operators with the inclusion of an adjustable parameter are defined on the basis of the proposed operational laws, and several instrumental properties are then investigated. Furthermore, some limiting cases of the proposed IVPFFP operators are analyzed with respect to the varying adjustable parameter values. Finally, an IVPFFP aggregation operator-based multiple attribute group decision-making model is developed with a practical example furnished to demonstrate its feasibility and efficiency. The power that the adjustable parameter exhibits has been leveraged to affect the final decision results, and the proposed IVPFFP operators are compared with three selected aggregation operators to demonstrate their advantages provided with a practical example.