Interval‐valued probabilistic linguistic term sets in multi‐criteria group decision making

The theory of probabilistic linguistic term sets (PLTSs) is very useful in objectively dealing with the multi‐criteria group decision making (MCGDM) problems in which there is hesitancy in providing linguistic assessments; and PLTSs allow experts to express their preferences on one linguistic term over another. In order to reflect the uncertainty and inconsistency of decision‐makers and handle incomplete linguistic information, we propose a new PLTS called interval‐valued probabilistic linguistic term set (IVPLTS). In addition, the existing approaches associated with PLTSs are limited or highly complex in real applications. Therefore, new operations, comparison laws, and aggregation operators are developed for IVPLTS. Furthermore, we establish an efficient framework for MCGDM problems based on the proposed comparison method and the fuzzy preference relation. Then we apply it to a real‐life case under linguistic environment. The extended TOPSIS methods combined with PLTSs by using different operational laws are also included for comparison. The final results demonstrate the efficiency and practicality of the new framework.     

Uncertainty measures for probabilistic hesitant fuzzy sets in multiple criteria decision making

This contribution reviews critically the existing entropy measures for probabilistic hesitant fuzzy sets (PHFSs), and demonstrates that these entropy measures fail to effectively distinguish a variety of different PHFSs in some cases. In the sequel, we develop a new axiomatic framework of entropy measures for probabilistic hesitant fuzzy elements (PHFEs) by considering two facets of uncertainty associated with PHFEs which are known as fuzziness and nonspecificity. Respect to each kind of uncertainty, a number of formulae are derived to permit flexible selection of PHFE entropy measures. Moreover, based on the proposed PHFE entropy measures, we introduce some entropy-based distance measures which are used in the portion of comparative analysis. Eventually, the proposed PHFE entropy measures and PHFE entropy-based distance measures are applied to decision making in the strategy initiatives where their reliability and effectiveness are verified.     

Signed distance‐based ORESTE for multicriteria group decision‐making with multigranular unbalanced hesitant fuzzy linguistic information

The objective of this study is to develop an integrated approach for solving multicriteria group decision‐making problems with multigranular unbalanced hesitant fuzzy linguistic term sets (HFLTSs). Firstly, a signed distance‐based transformation function is proposed to unify multigranular unbalanced hesitant fuzzy linguistic (HFL) assessments. Secondly, a mathematical programming model based on the maximum consensus is constructed to allocate decision‐makers (DMs)' weights objectively. Thirdly, a new signed distance‐based preference score function is defined to aggregate HFL assessments and determine the weak ranking of alternatives, and a novel preference, indifference, and incomparability test framework is constructed to identify the subtle relations among alternatives. On these bases, a signed distance‐based ORESTE (Organísation, rangement et Synthèse de données relarionnelles, in French) method, in which knowledge regarding criterion values and weights are expressed as multigranular unbalanced HFLTSs, is developed to obtain the ranking of alternatives. Finally, an illustrative example, followed by sensitivity and comparative analyses, is presented to verify the feasibility and effectiveness of the proposed approach.     

An outranking method for multicriteria decision making with probabilistic hesitant information

Defects of hesitant fuzzy set (HFS) manifest in actual decision-making process, so adding probabilities to the values in HFS is necessary. The probabilistic HFS (PHFS) is a useful tool to describe the uncertainty of elements in HFS by introducing occurrence probabilities. However, some important issues in PHFS utilization remain to be addressed. In this study, an outranking method for multicriteria decision making (MCDM) with probabilistic hesitant information is presented. First, the binary relations between two probabilistic hesitant fuzzy elements (PHFEs) are defined on the basis of the elimination and choice translating reality method. Some outranking relations between the alternatives are then introduced. Second, we provide a Hausdorff distance between two PHFEs. The main characteristic of the proposed Hausdorff distance is that it does not require the same length and arrangement of the PHFEs. Third, a maximizing Hausdorff distance deviation method is developed to obtain the evaluation criteria weights under a probabilistic hesitant fuzzy environment. Finally, an illustrative example in conjunction with comparative analysis is used to demonstrate that the proposed method is feasible for practical MCDM problems.     

New exponential operational laws and their aggregation operators for interval‐valued Pythagorean fuzzy multicriteria decision‐making

In this article, we define two new exponential operational laws about the interval‐valued Pythagorean fuzzy set (IVPFS) and their corresponding aggregation operators. However, the exponential parameters (weights) of all the existing operational laws of IVPFSs are crisp values in IVPFS decision‐making problems. As a supplement, this paper first introduces new exponential operational laws of IVPFS, where bases are crisp values or interval numbers and exponents are interval‐valued Pythagorean fuzzy numbers. The prominent characteristic of these proposed operations is studied. Based on these laws, we develop some new weighted aggregation operators, namely the interval‐valued Pythagorean fuzzy weighted exponential averaging operator and the dual interval‐valued Pythagorean fuzzy weighted exponential averaging. Finally, a decision‐making approach is presented based on these operators and illustrated with some numerical examples to validate the developed approach.     

Some methods for strategic decision‐making problems with immediate probabilities in Pythagorean fuzzy environment

In this article, a new decision‐making model with probabilistic information and using the concept of immediate probabilities has been developed to aggregate the information under the Pythagorean fuzzy set environment. In it, the existing probabilities have been modified by introducing the attitudinal character of the decision maker by using an ordered weighted average operator. Based on it, we have developed some new probabilistic aggregation operator with Pythagorean fuzzy information, namely probabilistic Pythagorean fuzzy weighted average operator, immediate probability Pythagorean fuzzy ordered weighted average operator, probabilistic Pythagorean fuzzy ordered weighted average, probabilistic Pythagorean fuzzy weighted geometric operator, immediate probability Pythagorean fuzzy ordered weighted geometric operator, probabilistic Pythagorean fuzzy ordered weighted geometric, etc. Furthermore, we extended these operators by taking interval‐valued Pythagorean fuzzy information and developed their corresponding aggregation operators. Few properties of these operators have also been investigated. Finally, an illustrative example about the selection of the optimal production strategy has been given to show the utility of the developed method.     

广义犹豫模糊信息集成及其多属性群决策

在犹豫模糊环境下,主要研究了基于阿基米德范数的广义信息集成算法,并提出了一种新的多属性群决策方法。基于阿基米德T-范数和S-范数,定义了广义犹豫模糊运算法则;运用新定义的广义犹豫模糊运算法则,提出了广义犹豫模糊有序加权平均(G-HFOWA)算子,研究了其优良性质;探讨了在某些特殊情况下,广义犹豫模糊有序加权平均算子将转化为一些常见的犹豫模糊信息集成算子,包括犹豫模糊有序加权平均算子、犹豫模糊Einstein有序加权平均算子、犹豫模糊Hamacher有序加权平均算子以及犹豫模糊Frank有序加权平均算子;基于广义信息集成算子,构建了一种新的犹豫模糊多属性群决策方法,并将其应用于区域经济协调发展研究过程中,以验证提出的决策方法是可行的与有效的。     

Multi-criteria group decision making with incomplete hesitant fuzzy preference relations

In order to simulate the hesitancy and uncertainty associated with impression or vagueness, a decision maker may give her/his judgments by means of hesitant fuzzy preference relations in the process of decision making. The study of their consistency becomes a very important aspect to avoid a misleading solution. This paper defines the concept of additive consistent hesitant fuzzy preference relations. The characterizations of additive consistent hesitant fuzzy preference relations are studied in detail. Owing to the limitations of the experts’ professional knowledge and experience, the provided preferences in a hesitant fuzzy preference relation are usually incomplete. Consequently, this paper introduces the concepts of incomplete hesitant fuzzy preference relation, acceptable incomplete hesitant fuzzy preference relation, and additive consistent incomplete hesitant fuzzy preference relation. Then, two estimation procedures are developed to estimate the missing information in an expert's incomplete hesitant fuzzy preference relation. The first procedure is used to construct an additive consistent hesitant fuzzy preference relation from the lowest possible number, (n − 1), of pairwise comparisons. The second one is designed for the estimation of missing elements of the acceptable incomplete hesitant fuzzy preference relations with more known judgments. Moreover, an algorithm is given to solve the multi-criteria group decision making problem with incomplete hesitant fuzzy preference relations. Finally, a numerical example is provided to illustrate the solution processes of the developed algorithm and to verify its effectiveness and practicality.     

The linear assignment method for multicriteria group decision making based on interval‐valued Pythagorean fuzzy Bonferroni mean

The interval‐valued Pythagorean fuzzy sets can easily handle uncertain information more flexibly in the process of decision making. Considering the interrelationship among the input arguments, we extend the Bonferroni mean and the geometric Bonferroni mean to the interval‐valued Pythagorean fuzzy environment and solve its practical application problems. First, we develop the interval‐valued Pythagorean fuzzy Bonferroni mean and the weighted interval‐valued Pythagorean fuzzy Bonferroni mean (WIVPFBM) operators. The properties of these aggregation operators are investigated. Then, we also develop the interval‐valued Pythagorean fuzzy geometric Bonferroni mean and the weighted interval‐valued Pythagorean fuzzy geometric Bonferroni mean (WIVPFGBM) operators and analyze their properties. Third, we utilize the WIVPFBM and WIVPFGBM operators to fuse the information in the interval‐valued Pythagorean fuzzy multicriteria group decision making (IVPFMCGDM) problem, which can obtain much more information in the process of group decision making. With the aid of the linear assignment method, we present its extension and further design a new algorithm for the application of IVPFMCGDM. Finally, an example is given to elaborate our proposed algorithm and validate its excellent performance.     

Expected hesitant VaR for tail decision making under probabilistic hesitant fuzzy environment

Generally, to reasonably make decision, all evaluation information should be aggregated, and thus, the ranking and the optimal alternative can be obtained. However, in some extreme cases, the decision maker (DM) can only focus on the tail information such as the big-loss or big-gain values and wants to ask the simple question “How bad can a thing become?” or “How good can a thing become?” To address this type of decision-making issue, this paper introduces the definition of value at risk (VaR), which is a famous term in the financial field, and the probabilistic hesitant fuzzy element (PHFE), which is a general hesitant fuzzy element (HFE) and has recently become a popular topic. Then, the hesitant VaR (HVaR) is defined, and its mathematical presentation is provided to measure the tail information of the PHFEs. It is found that the tail information calculated by the HVaR is segmentary, and only the boundary value is used. Therefore, this paper further develops the expected HVaR (EHVaR) to improve the HVaR, which can describe the entire tail information. Two simple examples are provided to show and compare the proposed HVaR and EHVaR. To apply the EHVaR into a group decision making that focuses on the tail information, this paper proposes a dynamic programming model to calculate the weights of the DMs based on the principle that the more accurate PHFE should be given a bigger weight. Then, the tail group decision making steps based on the EHVaR are presented. Finally, this paper provides an example of selecting the optimal stock for four newly listed stocks in China to demonstrate the effectiveness of the proposed approaches.     

Induced generalized hesitant fuzzy operators and their application to multiple attribute group decision making

In this paper, we develop a series of induced generalized aggregation operators for hesitant fuzzy or interval-valued hesitant fuzzy information, including induced generalized hesitant fuzzy ordered weighted averaging (IGHFOWA) operators, induced generalized hesitant fuzzy ordered weighted geometric (IGHFOWG) operators, induced generalized interval-valued hesitant fuzzy ordered weighted averaging (IGIVHFOWA) operators, and induced generalized interval-valued hesitant fuzzy ordered weighted geometric (IGIVHFOWG) operators. Next, we investigate their various properties and some of their special cases. Furthermore, some approaches based on the proposed operators are developed to solve multiple attribute group decision making (MAGDM) problems with hesitant fuzzy or interval-valued hesitant fuzzy information. Finally, some numerical examples are provided to illustrate the developed approaches.     

广义犹豫模糊Bonferroni平均及其多准则群决策方法

针对决策信息为犹豫模糊信息且输入的决策信息之间存在相互关系的多准则群决策问题,提出了一种基于阿基米德T-范数和S-范数的广义犹豫模糊Bonferroni平均（GHFBM）算子,并建立一种新的多准则群决策方法,其优点是不仅能够在信息集结的过程中考虑到输入变量之间的相互联系,还使得群决策方法应用范围更加广泛。在此过程中,研究了GHFBM算子的一些优良性质,包括单调性、幂等性、有界性以及置换不变性等。紧接着,分别对参数和加性算子赋予不同的数值和函数,详细探讨了GHFBM的一些常用形式。结合对计算机网络系统更新方案的选择实例,对提出的群决策方法进行合理性与有效性验证。     

Interval‐valued Pythagorean fuzzy GRA method for multiple‐attribute decision making with incomplete weight information

In this paper, the concept of multiple‐attribute group decision‐making (MAGDM) problems with interval‐valued Pythagorean fuzzy information is developed, in which the attribute values are interval‐valued Pythagorean fuzzy numbers and the information about the attribute weight is incomplete. Since the concept of interval‐valued Pythagorean fuzzy sets is the generalization of interval‐valued intuitionistic fuzzy set. Thus, due the this motivation in this paper, the concept of interval‐valued Pythagorean fuzzy Choquet integral average (IVPFCIA) operator is introduced by generalizing the concept of interval‐valued intuitionistic fuzzy Choquet integral average operator. To illustrate the developed operator, a numerical example is also investigated. Extended the concept of traditional GRA method, a new extension of GRA method based on interval‐valued Pythagorean fuzzy information is introduced. First, utilize IVPFCIA operator to aggregate all the interval‐valued Pythagorean fuzzy decision matrices. Then, an optimization model based on the basic ideal of traditional grey relational analysis (GRA) method is established, to get the weight vector of the attributes. Based on the traditional GRA method, calculation steps for solving interval‐valued Pythagorean fuzzy MAGDM problems with incompletely known weight information are given. The degree of grey relation between every alternative and positive‐ideal solution and negative‐ideal solution is calculated. To determine the ranking order of all alternatives, a relative relational degree is defined by calculating the degree of grey relation to both the positive‐ideal solution and negative ideal solution simultaneously. Finally, to illustrate the developed approach a numerical example is to demonstrate its practicality and effectiveness.     

Algorithms for complex interval‐valued q‐rung orthopair fuzzy sets in decision making based on aggregation operators,AHP, and TOPSIS

The interval‐valued q‐rung orthopair fuzzy set (IVq‐ROFS) and complex fuzzy set (CFS) are two generalizations of the fuzzy set (FS) to cope with uncertain information in real decision making problems. The aim of the present work is to develop the concept of complex interval‐valued q‐rung orthopair fuzzy set (CIVq‐ROFS) as a generalization of interval‐valued complex fuzzy set (IVCFS) and q‐rung orthopair fuzzy set (q‐ROFS), which can better express the time‐periodic problems and two‐dimensional information in a single set. In this article not only basic properties of CIVq‐ROFSs are discussed but also averaging aggregation operator (AAO) and geometric aggregation operator (GAO) with some desirable properties and operations on CIVq‐ROFSs are discussed. The proposed operations are the extension of the operations of IVq‐ROFS, q‐ROFS, interval‐valued Pythagorean fuzzy, Pythagorean fuzzy (PF), interval‐valued intuitionistic fuzzy, intuitionistic fuzzy, complex q‐ROFS, complex PF, and complex intuitionistic fuzzy theories. Further, the Analytic hierarchy process (AHP) and technique for order preference by similarity to ideal solution (TOPSIS) method are also examine based on CIVq‐ROFS to explore the reliability and proficiency of the work. Moreover, we discussed the advantages of CIVq‐ROFS and showed that the concepts of IVCFS and q‐ROFS are the special cases of CIVq‐ROFS. Moreover, the flexibility of proposed averaging aggregation operator and geometric aggregation operator in a multi‐attribute decision making (MADM) problem are also discussed. Finally, a comparative study of CIVq‐ROFSs with pre‐existing work is discussed in detail.     

广义犹豫正态模糊信息集成及其多属性群决策

定义了犹豫正态模糊元及其运算法则、得分函数、Euclidean距离等概念;提出了广义犹豫正态模糊有序加权平均算子,并研究其性质,该算子不仅尽可能多地保留决策者的偏好信息,还可依据决策者的主观意愿选择不同的参数和属性权重,使得决策结果达到决策者的期望值;紧接着对属性权重和算子参数赋予不同的数值,获取广义犹豫正态模糊有序加权平均算子的若干种特殊算子,并探讨两个常用算子的大小关系;针对属性权重完全未知的多属性群决策问题,构建一种基于广义犹豫正态模糊有序加权平均算子的群决策方法。该方法利用同一属性下所有方案属性值间的距离求得最优权重,然后将同一方案下各属性值集结成为综合属性值,进而得到方案优劣排序。通过实例分析说明该方法的可行性和有效性。     

基于区间梯形二型犹豫模糊数的多准则决策方法

针对准则权重不完全的犹豫模糊多准则决策问题,提出基于区间梯形二型犹豫模糊数的决策方法.首先,给出区间梯形二型犹豫模糊数,根据几何面积法定义区间梯形二型犹豫模糊数的可能度和差异度;然后,利用差异度和离差最大化模型得到各准则权重,基于TOPSIS思想得到各方案的综合贴近度,并对方案进行排序;最后,通过算例分析和对比分析验证了所提出方法的可行性和有效性.     

Uncertain linguistic hesitant fuzzy sets and their application in multi‐attribute decision making

To denote the quantitative and qualitative fuzzy information simultaneously, this paper introduces a new type of fuzzy sets called uncertain linguistic hesitant fuzzy sets, which are denoted by an uncertain linguistic variable with several possible interval membership degrees. Considering the application of this type of fuzzy sets, several basic operational laws are defined, and several properties are studied. Meanwhile, an ordered relationship is introduced. Then, two types of uncertain linguistic hesitant fuzzy aggregation operators are defined. One uses additive measures, and the other is based on λ‐fuzzy measures. Then, a similarity measure is presented, by which models for the optimal weight vector are constructed. After that, an approach to uncertain linguistic hesitant fuzzy multi‐attribute decision making is developed. Finally, an illustrative example for evaluating corporate environmental performance is offered to show the concrete practicality of the procedure.     

A new multiple criteria decision making method based on intuitionistic fuzzy information

The risk attitude of a decision maker is considered in the decision process. Inspired by mean-variance type utility functions in the financial risk management, a new class of decision functions are defined based on the weighted score function and the weighted accuracy function in the intuitionistic fuzzy setting. By choosing a suitable parameter value, the decision maker’s risk attitude can be flexibly reflected by our decision function. The new method can be applied for both the exactly known and partly known criteria weight situations. For the latter case, it is only necessary to solve one linear programming problem. The developed models and algorithms are then extended to multiple criteria decision making problems with the interval-valued intuitionistic fuzzy information. Numerical examples are provided to illustrate the practicality, flexibility and efficiency of our new algorithms.     

An extended COPRAS method for multiattribute group decision making based on dual hesitant fuzzy Maclaurin symmetric mean

The well-known Maclaurin symmetric mean (MSM) and the dual MSM (DMSM) are introduced as important operators to handle multiattribute group decision making (MAGDM) information. The MSM and the DMSM operators have the prominent characteristic of accurately describing the interdependence of multi-input arguments. Due to their advantage, we extend the MSM and the DMSM into the dual hesitant fuzzy environment to aggregate uncertain information. Particularly, we propose some novel aggregation operators, namely dual hesitant fuzzy MSM, weighted dual hesitant fuzzy MSM, dual hesitant fuzzy dual MSM, and weighted dual hesitant fuzzy dual MSM operators. Moreover, we study some properties and special remarks regarding different values of the parameter. With an extension of the complex proportional assessment method, we formulate a new approach for the dual hesitant fuzzy MAGDM. Finally, we test the applicability and feasibility of our proposed method by solving a mobile payment platform selection problem in Ghana.