High-dimensional and incomplete (HDI) data are involved frequently in big data-related industrial applications. Latent factor (LF) analysis aims at extracting the knowledge of great value from such extremely sparse HDI data efficiently. Non-negative LF models based on the single LF-dependent, non-negative, and multiplicative update rules exactly are the representative of LF analysis. However, these models face low generalization dilemma due to incompatible with general unconstrained optimization techniques. To address this issue, this article proposes a novel momentum-accelerated and biased unconstrained non-negative latent factor (MBUNLF) model, which matches with unconstrained optimization techniques. The proposed MBUNLF model is built on three main ideas: (a) Improving the generalization through a non-negative mapping function; (b) Capturing information among different entities through linear biases; (c) Accelerating convergence during the training process through generalized momentum method. Empirical studies on six datasets from industrial applications indicate that the proposed MBUNLF model outperforms nine state-of-the-art models when processing HDI data, reducing the root mean square error by 19.47% on average. It demonstrates the validity of the MBUNLF model in extracting non-negative LFs from HDI data.

High-Dimensional and Incomplete (HDI) tensors contain a wealth of knowledge and patterns, which are typically utilized to characterize complex relationships between entities in a variety of industrial applications. Currently, the neural network-based tensor factorization model has shown superiority when handling the missing data in HDI tensors. However, it only uses the outer product of latent factors (LFs) of entities and neglects the interactions between the LF. In addition, the simple linear operation does not consider the nonlinear structure of the HDI tensor. To overcome the aforementioned issues, an Attention-mechanism-based Neural Latent-Factorization-of-Tensors (ANLFT) model is provided in this article. It encompasses three primary ideas: (a) incorporating the theory of neural networks with the latent factorization of tensor to construct the nonlinear structure in the HDI tensor effectively; (b) adopting the attention mechanism to depict the interactions between LF; (c) using the position-transitional particle swarm optimization backward propagation learning ( \(\rm{P}^{2}\) BP) scheme to train the ANLFT model efficiently. The experimental results on eight HDI datasets show that the ANLFT model can obtain higher estimation performance gain than state-of-the-art models. The convergence performance of the proposed model is also competitive with that of state-of-the-art models.

Plenty of online customer reviews (OCRs) provide consumers with an abundant source of word‐of‐mouth (WOM). It makes potential customers evaluate alternative products or services more conveniently. Thus, relative studies have been arising. Considering that existing research is almost deployed from the perspective of reviewers, the study proposes an evaluation model based on the OCRs from the view of potential customers. In the model, the OCRs in text form are translated into probabilistic linguistic term sets from the perspective of the cognitive process model, namely, the elaboration likelihood model, for further evaluation. Meanwhile, the influence of the OCRs’ published time is also taken into consideration, and then, the model is transformed into a dynamic evaluation model. In general, we propose a dynamic product evaluation model to simulate the cognitive process of the potential consumers disposing of OCRs. In addition, an evaluation of 6 hotels in Chengdu, Sichuan Province, is developed based on the proposed dynamic product evaluation model, and some discussions, as well as conclusions, are also carried out.

A dual hesitant fuzzy set (DHFS) describes the uncertainty in the real world by using the membership degree and nonmembership degree. It can collect fuzzy information comprehensively and apply them into decision-making tasks efficiently. In this article, we extract some characteristics, such as the average function, variance function, hesitancy degree to describe a dual hesitant fuzzy element, and develop novel distance measures of DHFSs based on these characteristics. Further, we investigate their properties and prove the triangle inequality of distance measure. Finally, we apply it in practical medical diagnosis to illustrate the validity of our proposed distance measures.

Distance measure is an essential tool to characterize the difference between two samples. Recently, lots of distance measures have been proposed for hesitant fuzzy sets (HFSs). In this paper, we shall propose some novel distance formulas to measure the deviation between two HFSs. First, we define some new concepts including the hesitant fuzzy variance, covariance, and correlation coefficient. Based on these concepts and the idea of the traditional Mahalanobis distance, the hesitant Mahalanobis distance between two HFSs is developed. Then we discuss the properties of the new distance measure and uncover the significant characteristic of the introduced distance measure that it can give the attributes an adaptive weight and can eliminate the influence of the correlation between the attributes under hesitant fuzzy environment. And then, some extensions of this new distance measure are also developed. Second, to show the validity and applicability of the proposed distance measures, we compare them with the existing ones in decision making and cluster analysis with some numerical examples. Third, using the proposed distance measures, we develop two algorithms to estimate the optimal number of clusters, which is a new application area of the hesitant fuzzy distance measures. Finally, the two algorithms are applied in a numerical example to illustrate their applicability and efficiency.

For the application of probabilistic linguistic term sets (PLTSs) in multicriteria group decision making (MCGDM), this paper aims to develop an approach to solve the preference problem. First, a belief interval interpretation of PLTSs is presented, which makes it possible to represent the mathematical operations on PLTSs as the operations on belief intervals. This can reduce the uncertainty degree of information caused by the differentiated knowledge and cognitions of decision makers. Then, two methods of belief interval measure are proposed. One is the distance measure, which is utilized to obtain the criteria weights of each decision maker, so as to recognize the preferences for criteria. The other is the probability degree, which is used to get the partial order relation for alternatives of each decision maker, and to recognize the preferences for alternatives. Next, employing Dempster's rule of combination and graph theory, a visual algorithm is constructed to solve the MCGDM preference problem in the application of PLTSs. Finally, an illustrative example for the selection of emergency materials deployment scheme and the comparative analyses are shown to demonstrate the effectiveness of the proposed method.

Double hierarchy linguistic term set (DHLTS) is a powerful tool when expressing the real thoughts of experts and handling complex linguistic information considering that it divides complex linguistic information into two simple linguistic hierarchies in which the first hierarchy linguistic term set (LTS) is the main linguistic hierarchy and the second hierarchy LTS is the linguistic feature or detailed supplementary of each linguistic term in the first hierarchy LTS. Some extensions of DHLTS have been developed, such as the double hierarchy hesitant fuzzy LTS, the unbalanced DHLTS, the linguistic preference ordering, the double hierarchy linguistic preference relation, and the double hierarchy hesitant fuzzy linguistic preference relation. In recent years, DHLTS and its extensions have been researched by scholars in lots of fields, including the extended concepts, the operational laws, the comparative methods, the measure methods, the consistency and consensus methods, the decision-making methods, the applications, and so forth. Therefore, the purpose of this paper is to review all the researches of DHLTS and its extensions, as well as proposing some challenges in the future.

This paper proposes an approach to linguistic three-way decision making problem with double hierarchy linguistic term evaluation information. Double hierarchy linguistic term set consists of the first hierarchy and second hierarchy linguistic term set, which can describe uncertainty and fuzziness more flexibly. First, the Hamacher operational rules, score function and distance measure of double hierarchy linguistic elements are defined. Next, we construct the double hierarchy linguistic decision-theoretic rough set model. And the conditional probability is calculated based on the double hierarchy linguistic term environment with grey relational analysis, which makes the process of decisions more rational. Then the loss functions are aggregated by the double hierarchy linguistic Hamacher weighted averaging operator, which takes into account different decision attitudes of decision makers. And the results of decision are deduced by the minimum-loss principle. Finally, a case study about the selection of cooperation companies during the COVID-19 is used to demonstrate the practicability of our proposed method.

Multi-attribute decision-making (MADM) is one of the most important parts in decision-making theory, and related research is becoming more and more popular over the past few years. Investigating that the information could be qualitative and quantitative, and the different measurement units cause difficulties in some MADM problems, the additive ratio assessment system (ARAS) method was proposed. The method tries to solve MADM problems through a simple way efficiently, and at the same time eliminates the influence of different measurement units. Till now, the method has received extensive attention and has been extended to different information environments and application fields. To know about the development of the method and improve the method efficiently, this paper reviews the studies on the ARAS method from the perspectives of basic information (including the bibliometrics analyses and the outline of ARAS method), the development on theory (including the development on MADM mechanism, different information environments, and combination with different methods), the development on the application and the future challenge. From the overview, the basic situations and the development of the ARAS method are presented clearly, and the analyses of the challenges can also provide useful and sufficient instructions for the future application and improvement of the method.

In recent decades, image segmentation has aroused great interest of many researchers, and has become an important part of machine learning, pattern recognition, and computer vision. Among many methods of image segmentation, fuzzy C-means (FCM) algorithm is undoubtedly a milestone in unsupervised method. With the further study of FCM, various different kinds of FCM algorithms are put forward to deal with the specific problems in image segmentation. Because there exist uncertainties in different regions of the image and similarity in the same region, reducing the uncertainty is still the main problem in image segmentation. Considering that Pythagorean fuzzy set (PFS) is a powerful tool to deal with uncertainty, in this paper, we use PFS to describe the uncertainty of image segmentation, including introducing fuzzification and defuzzification process and Pythagorean fuzzy element to describe the membership degree of pixel, combine the neighborhood information with weights and Pythagorean fuzzy distance, and propose Pythagorean fuzzy C-means (PFCM) algorithm. Finally, we apply PFCM algorithm in image segmentation, such as different size images and Berkeley Segmentation Data Set to illustrate the effectiveness and applicability of our proposed algorithm. Meanwhile, we do comparison analysis between PFCM, fully convolution network and Deep-image-Prior networks, these results show that our proposed PFCM has good intuition and effectiveness.

A measure is necessary for various decision-making problems, but it is inherently difficult for fuzzy information. This paper defines an extended probabilistic hesitant I-fuzzy set (PHIFS), and develops a simple and effective measure based on weighted area index. The measure can consider players' risk preferences and it has appealing properties such as linearity. Then it is applied in multiattribute decision making and game decision making (GDM) without difficulty, and the validity and simplicity are shown by comparing the performances. Especially, the optimal strategies of GDM with payoffs of PHIFSs are obtained by the parameterized linear programming models, and the results shows that the measure is flexible and practical in interactive decision making.

In multiattribute decision making, the analytic network process (ANP) is an important methodology to derive the subjective weights of attributes when the dependence and feedback relations exist between attributes, and the number of attributes should be no more than seven in a comparison matrix. To reduce the dimensions of attributes, we propose a hybrid hesitant fuzzy linguistic factor analysis method to cluster the attributes into main factors. The method takes multiple forms of decision-making information into consideration, such as single linguistic terms, hesitant fuzzy linguistic terms, and numeric values. Meanwhile, the objective weights of the main factors are obtained as well. As for the subjective weights of main factors, the incomplete probabilistic linguistic ANP is developed after improving the incomplete probabilistic linguistic preference relation with multiplicative consistency. At last, the final weights of the main factors are calculated by combining the objective and subjective weights. A questionnaire survey about assessing the weights of the main factors influencing graduate students' physical health is designed to explain the application of the proposed methodology. To sum up, the main importance and contributions of this study are as following: (1) developing a hybrid hesitant fuzzy linguistic factor analysis method and incomplete probabilistic linguistic ANP, (2) proposing a novel weight-derived method from both objective and subjective perspectives, and (3) applying it to graduate students' physical health assessment.

The notion of intuitionistic fuzzy soft sets (IFSSs) provides an effective tool for solving multiple attribute decision making with intuitionistic fuzzy information. The most crucial issue in decision making based on IFSSs is how to derive the ranking of alternatives from the information quantified in terms of intuitionistic fuzzy values. In this study, we propose a new extension of the preference ranking organization method for enrichment evaluation (PROMETHEE), by taking advantage of IFSSs. In addition to presenting a myriad of new notions, such as intuitionistic fuzzy membership (or nonmembership) deviation matrices, intuitionistic fuzzy membership (or nonmembership) preference matrices, and aggregated intuitionistic fuzzy preference matrices, we put more emphasis on the construction of three distinct preference structures and related utility functions on the corresponding weakly ordered sets by considering the positive, negative, and net flows of the alternatives based on the aggregated intuitionistic fuzzy preference matrix. We present a new algorithm for solving multiple attribute decision-making problems with the extended PROMETHEE method based on IFSSs. Moreover, a benchmark problem concerning risk investment is investigated to give a comparative analysis and show the feasibility of our approach.

The probabilistic hesitant fuzzy preference relations (PHFPRs) provide the decision makers with an efficient means to express the preference information on pairwise comparisons over alternatives. In this paper, we propose an automatic consistency improving model for PHFPRs. First, we propose a novel normalization algorithm to normalize probabilistic hesitant fuzzy elements (PHFEs) by using probability splitting idea and develop novel operational laws. Then, we define the consistency index to compute the degree of deviation between the PHFPRs and their multiplicative consistent PHFPRs. We also develop a novel consistency threshold estimation method for obtaining the threshold of consistency index and then put forward an automatic consistency improving algorithm for repairing inconsistent PHFPRs. Moreover, two probabilistic hesitant fuzzy aggregation operators are put forward to aggregate preference values in acceptably multiplicative consistent PHFPRs for obtaining the ranking orders of alternatives. Finally, an illustrative example is given to show the implementation process of our proposed automatic consistency improving model and also we compare our proposed model with the existing studies.

The probabilistic linguistic term set is a flexible and efficient tool to represent the cognitive complex information of experts. It has attracted many scholars’ attention since it was proposed. Information fusion over the cognitive complex information is a significant issue for decision-making problems. Over the past years, more than 40 aggregation operators have been proposed to fuse the probabilistic linguistic term sets. The aim of this paper is to survey the existing probabilistic linguistic aggregation operators from the perspectives of principles, definitions, classifications, and applications. To do so, first, we summarize the present normalization techniques and operations of probabilistic linguistic term sets. Afterward, this study classifies the existing probabilistic linguistic aggregation operators into 12 kinds. Then, the application areas of these probabilistic linguistic aggregation operators are outlined. Future research directions with interests are proposed to tackle present challenges.

The q-rung orthopair fuzzy set is characterized by membership and nonmembership functions, and the sum of the qth power of them is less than or equal to one. Since it releases the constraints existed in both intuitionistic fuzzy set and Pythagorean fuzzy set, it has wide applications in real cases. However, so far, there is little research on the multiplicative consistency of q-rung orthopair fuzzy preference relation (q-ROFPR). To fill this vacancy, this paper provides a detailed analysis on the multiplicative consistency of q-ROFPR. First, we investigate the concept of multiplicative consistent q-ROFPR and its properties. Subsequently, two goal programming models are proposed to derive the priorities from individual and group q-ROFPRs, respectively. After that, a novel consistency-improving algorithm for q-ROFPR and a weight-generating method for decision-makers are discussed in detail, based on which, a novel group decision-making method is proposed. Finally, a case study concerning the evaluation of rehabilitation program selection is given to illustrate the applicability of the proposed method. The effectiveness and superiority of the proposed method are verified by comparing it with some existing methods.

The induced ordered weighted average is an averaging aggregation operator that provides a parameterized family of aggregation operators between the minimum and the maximum. This paper presents some new generalizations by using Bonferroni means (BM) forming induced BM. The main advantage of this approach is the possibility of reordering the results according to complex ranking processes based on order-inducing variables. The work also presents some additional extensions by using the weighted ordered weighted average, immediate weights, and hybrid averages. Some further generalizations with generalized and quasi-arithmetic means are also developed to consider a wide range of particular cases including quadratic and geometric aggregations. The article also considers the applicability of the new approach in-group decision-making developing an application in sales forecasting.

The interval-valued q-rung orthopair fuzzy set (IVq-ROFS) provides an extension of Yager's q-rung orthopair fuzzy set (q-ROFS), where membership and nonmembership degrees are subsets of the closed interval [ 0 , 1 ] . In such a situation, it is more superior for decision makers to provide their judgments by intervals instead of crisp numbers due to the uncertainty and vagueness in real life. In this paper, we study the calculus theories of IVq-ROFS from the microscopic. In particular, we first introduce the elementary arithmetic of interval-valued q-rung orthopair fuzzy values (IVq-ROFVs), including addition, multiplication, and their inverse. They are the basis for analysis and calculation throughout the work. In addition, we discuss and prove in detail the operation properties and aggregation operators of IVq-ROFVs. Then, we introduce the concept of interval-valued q-rung orthopair fuzzy functions (IVq-ROFFs), which is the main research object of this paper. After that, we further discuss the continuity, derivatives and differentials of IVq-ROFFs. We also find that the derivatives of IVq-ROFFs are closely related to elasticity, which is an important concept in economics. Finally, we provide some application examples to verify the feasibility and effectiveness of the derived results.

We consider the multicriteria decision-making (MCDM) problems where there exists a prioritization relationship over the criteria. We introduce the concept of the priority degree. Then we give three kinds of prioritized aggregation operators based on the priority degrees: the prioritized averaging operator with the priority degrees, the prioritized scoring operator with the priority degrees, and the prioritized ordered weighted averaging operator with the priority degrees. Some desired properties of these prioritized aggregation operators are also investigated. The priority degree plays an important role in the prioritized MCDM problems. We also investigate how to select a proper priority degree according to the giving decision information. By using an illustrative example, we show that the prioritized aggregation operators based on the priority degrees provide the decision-makers more choices and they are more flexible in the process of decision-making.

Intuitionistic fuzzy multiplicative preference relations (IFMPRs), as an extension of multiplicative preference relations, can denote the decision-makers’ (DMs’) preferred and nonpreferred degrees simultaneously. Just as any other type of preference relations, consistency is crucial to guarantee the rational ranking orders. Thus, this paper introduces a new consistent concept for IFMPRs that is a natural extension of crisp case and overcomes the issues in the previous concepts of consistency. To judge the consistency of IFMPRs, several programming models are constructed, and an approach to deriving completely consistent IFMPRs is presented. Considering incomplete case, consistency-based models are built to determine missing values that can address incomplete IFMPRs with the ignored objects, namely, all information for them is unknown. After that, group decision-making with IFMPRs is studied. To measure the agreement degree between the DMs’ individual IFMPRs, a new consensus index is defined, and an interactive algorithm to improve the consensus is offered. Based on the consistency and consensus analysis, a new method to group decision-making with IFMPRs is developed. Finally, case studies are offered to show the application of the new procedure and to compare it with previous methods.

The ordered weighted averaging (OWA) operators play a crucial role in aggregating multiple criteria evaluations into an overall assessment supporting the decision makers’ choice. One key point steps is to determine the associated weights. In this paper, we first briefly review some main methods for determining the weights by using distribution functions. Then we propose a new approach for determining OWA weights by using the regular increasing monotone quantifier. Motivated by the idea of normal distribution-based method to determine the OWA weights, we develop a method based on elliptical distributions for determining the OWA weights, and some of its desirable properties have been investigated.

The probabilistic hesitant fuzzy set (PHFS) associates the probability with the hesitant fuzzy set (HFS), which has been proposed to improve the granularity of the HFS and can remain more information, is significant to solve the multicriteria group decision-making (MCGDM) problems when the decision makers fail to provide their preferences completely. To express the probability information existing in the hesitancy more conveniently, we propose a generalized form of P-HFS named interval-valued probabilistic hesitant fuzzy set (IVPHFS). In addition, we define some basic operation laws and aggregation operators of IVPHFSs. Based on which, we provide an efficient approach to deal with the practical MCGDM problems by IVPHFSs aggregation operators under the interval-valued probabilistic hesitant fuzzy environment. Last but not least, we apply the proposed approach to the research of the Arctic geopolitical risk evaluation. The method based on the score function of the probabilistic dual hesitant fuzzy set is also introduced for comparison. The comparing results demonstrate that our approach is more reasonable and logical.

The hesitant Pythagorean fuzzy set is frequently considered as a solution for decision making under uncertainty. Whereas the representation of uncertain information might be not sufficient in the hesitant Pythagorean fuzzy environment, thus the concept of interval-valued hesitant Pythagorean fuzzy sets (IVHPFSs) is proposed. Specifically, we first propose the concept of IVHPFSs and then study the operational rules and distance measures of IVHPFSs in detail. To ease the possible application, we explore two decision-making processes in the setting of IVHPFSs by drawing support from the technique for order preference by similarity to ideal solution and Choquet integral-based method. Finally, the selecting processes of project private partner are also presented to demonstrate the decision-making processes based on IVHPFSs and compared with some similar techniques.

The q-rung orthopair fuzzy set ( q-ROFS) that the sum of the qth power of the membership degree and the qth power of the nonmembership degree is restricted to one is a generalization of fuzzy set (FS). Recently, many researchers have given a series of aggregation operators to fuse q-rung orthopair fuzzy discrete information. Subsequently, although some scholars have also focused on studying q-rung orthopair fuzzy continuous information and give its continuity, derivative, differential, and integral, those studies are only considered from the perspective of multivariable fuzzy functions. Thus, the main aim of the paper is to study the q-rung orthopair fuzzy continuous single variable information. In this paper, we first define the concept of q-rung orthopair single variable fuzzy function ( q-ROSVFF) to describe the fuzzy continuous information, and give its domain to make sure that this kind of function is meaningful. Afterward, we propose the limits, continuities, and infinitesimal of q-ROSVFFs, and offer the relationship between the limit of q-ROSVFF and that of q-ROSVFF infinitesimal. On the basis of the definition of derivative in mathematical analysis, we define the subtraction and division derivatives and basic operational rules, and offer the simpler proofs for the derivatives of q-ROSVFFs. What is more, we propose the subtraction and division differential invariances, and give the approximate calculation formulas of q-ROSVFFs when the value of independent variable is changed small enough. In the real situation, fundamental functions cannot be used to express more complicated functions, thus we define the compound q-ROSVFFs and give their chain rules of subtraction and division derivatives. Finally, we use numerical examples by simulation to verify the feasibility and veracity of the approximate calculation on q-ROSVFFs.

Qualitative portfolio selection approach is a suitable technique for obtaining an optimal portfolio when quantitative data are unavailable and traditional portfolio models are ineffective. However, few studies focus on this issue. This study addresses the lack of research by defining the score-hesitation trade-off rule and introducing the intuitionistic fuzzy set (IFS), based on which an intuitionistic fuzzy portfolio selection (IFPS) model is proposed. The IFS is introduced because of its comprehensive consideration of preference and nonpreference, and is used to represent qualitatively evaluated information from investors and experts. Furthermore, an intuitionistic fuzzy investment scenario is established and a trisection approach is designed to distinguish three types of risk investors, based on which three corresponding IFPS models are constructed. After this, a portfolio selection process under the intuitionistic fuzzy environment is provided, and a simple example is given to show the application of the process. In addition, the investment opportunities and efficient frontier of the IFPS model are investigated to demonstrate the effectiveness of the proposed portfolio selection model. Finally, an example of calculating optimal investment ratios and selecting an optimal portfolio for four newly listed stocks in China is provided to demonstrate the feasibility and practicability of the proposed approaches.

Owing to the information insufficiency, it might be difficult for decision makers to precisely evaluate their assessments in real decision-making. As a new extension of the Pythagorean fuzzy sets, the interval-valued Pythagorean fuzzy sets (IVPFSs) can availably provide enough input space for decision makers to evaluate their assessments with interval numbers. By extending the Bonferroni mean to model the heterogeneous interrelationship among attributes, the extended Bonferroni mean (EBM) was examined. Considering the partition structure of relationship among the attributes, we introduce the EBM into the interval-valued Pythagorean fuzzy environment and develop two new aggregation operators, namely, interval-valued Pythagorean fuzzy extended Bonferroni mean and weighted interval-valued Pythagorean fuzzy extended Bonferroni mean (WIVPFEBM) operators. Meanwhile, some of their special cases and properties are also deeply discussed. Subsequently, by employing the WIVPFEBM operator, we propose an approach for multiple attribute decision making with IVPFSs. Finally, a practical illustration of the E-commerce project selection problem is investigated by our proposed method, which successfully demonstrates the applicability of our results.

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.

In this paper, we study the well-known Bonferroni mean and develop its generalized aggregation operators in the Pythagorean fuzzy environment. More specifically, by considering the interrelationship between arguments with Pythagorean fuzzy information, we develop the Pythagorean fuzzy Bonferroni mean (PFBM) and some special properties and cases of them are also discussed. Furthermore, taking the multicriteria decision making environment into consideration, we extend the results of PFBM and develop the weighted Pythagorean fuzzy Bonferroni mean (WPFBM). Meanwhile, we also propose an approach for the application of WPFBM. However, during the application of the WPFBM operator, the calculation is very complex and time consuming. Hence, we introduce the multithreading into the application of the WPFBM operator and develop an accelerative calculating algorithm for it. To validate the performance of the accelerative calculating algorithm, we further design the corresponding experimental analysis.

As a novel and powerful tool, the notion of probabilistic linguistic term sets (PLTSs) can efficiently model this kind of qualitative assessment information utilizing several possible linguistic terms associated with probabilities or weights over alternatives. Considering that there are no investigation and research on the correlation coefficient and clustering analysis for the concept of PLTSs. Therefore, some correlation coefficient formulas are put forward to measure the relationship between two PLTSs and then they are utilized to develop two novel clustering algorithms to group PLTSs in this paper. We first define some correlation coefficient formulas and their weighted forms to measure the relationship between PLTSs. Then, we extend a fuzzy clustering algorithm for PLTSs and also propose a novel orthogonal clustering algorithm for PLTSs. Finally, we provide a practical example, which performs cluster analysis on the levels of general higher education in different regions of China, to test and verify the usability of our proposed clustering algorithm.

The uncertainty and complexity of the decision-making environment and the subjectivity of the decision makers will lead to the inevitable errors of the decision-making data. A poor decision will be produced with those errors, whereas the linear programming technique for multidimensional analysis of preference (LINMAP) method can adjust such errors through constructing an optimal programming model based on the consistency of the decision-making information, and it has been applied widely in multiple attribute group decision making (MAGDM). Moreover, Pythagorean fuzzy information is useful to simulate the ambiguous and uncertain decision-making environment. Therefore, the LINMAP method under the Pythagorean fuzzy circumstance will be proposed in this paper to solve MAGDM problems. To measure the fuzziness and uncertainty of Pythagorean fuzzy set (PFS) and interval-valued PFS, Pythagorean fuzzy entropy (PFE) and interval-valued PFE (IVPFE) grounded on the similarity and hesitancy parts have been defined, respectively. Then, Pythagorean fuzzy LINMAP (PF LINMAP) methods are constructed on the basis of the PFE and IVPFE correspondingly. Under the given preference relations, the maximum consistency and the amount of knowledge can be realized by the proposed methods. After investigating the relevant indicator system, the decision-making problem concerning railway project investment has been solved through the proposed PF LINMAP method with PFE. Finally, the practicability and effectiveness of the PF LINMAP method has been verified via the comparative analysis with the existing methods.

On the basis of the hesitant fuzzy membership, this study proposes the extended intuitionistic fuzzy set (EIFS) and the extended intuitionistic fuzzy number (EIFN) to synthesize the characters of the intuitionistic fuzzy set and the hesitant fuzzy set. We further develop two simplified and applied EIFSs, namely the credible EIFS (C-EIFS) and the possible EIFS (P-EIFS), to comprehensively mine the hesitant fuzzy membership information and to avoid the logical difficulty of simultaneously providing the membership and non-membership in each EIFS or EIFN. Then we investigate the foundations of C-EIFS and P-EIFS, including their expressions, operations, functions, differences, and selection rules. The corresponding aggregation operators are also proposed, and the calculation and relationships of these operators are proven. The prominent properties of C-EIFNs and P-EIFNs are focused on the boundary and average values, respectively; that is, the C-EIFN tends to aggregate the extreme information, whereas the P-EIFN prefers aggregating complete information. Therefore, applying them to decision making with risk preference is suitable, and two risk preference investment cases are provided to demonstrate the applications of these concepts and approaches.

As a new generalization of fuzzy sets, Pythagorean fuzzy sets (PFSs) can availably handle uncertain information more flexibly in the process of decision making. Through synthesizing the Bonferroni mean and the geometric mean, the geometric Bonferroni mean (GBM) captures the interrelationship of the input arguments. Considering the interrelationship among the input arguments, we introduce GBM into Pythagorean fuzzy situations and expand its applied fields. Under the Pythagorean fuzzy environment, we develop the Pythagorean fuzzy geometric Bonferroni mean and weighted Pythagorean fuzzy geometric Bonferroni mean (WPFGBM) operators describing the interrelationship between arguments and some special properties of them are also investigated. Then, we employ the WPFGBM operator to fuse the information in the Pythagorean fuzzy multicriteria group decision making (PFMCGDM) problem, which can obtain much more information in the process of group decision making. With the aid of the projection model, we present its extension and further design a new method for the application of PFMCGDM. Finally, an example is given to elaborate on the performance of our proposed method.

Since the Atanassov's intuitionistic fuzzy set theory was introduced, many intuitionistic fuzzy aggregation operators have been proposed for the integration of the intuitionistic fuzzy information. It is necessary to extend them to accommodate the infinite situations so that we can deal with the large amount of intuitionistic fuzzy data, and, at the same time, we must solve an important issue that whether the sum of an infinite number of intuitionistic fuzzy numbers is convergent or not. Thus, in this paper, we first put forward the infinite intuitionistic fuzzy series and product. Then, we investigate the properties and the convergences of the infinite intuitionistic fuzzy series and product, respectively. These results greatly enrich the intuitionistic fuzzy calculus theory.

In practical decision-making processes, we can utilize various types of fuzzy sets to express the uncertain and ambiguous information. However, we may encounter such the situations: the sum of the support (membership) degree and the against (nonmembership) degree to which an alternative satisfies a criterion provided by the decision maker may be bigger than 1 but their square sum is equal to or less than 1. The Pythagorean fuzzy sets (PFS), as the generalization of the fuzzy sets, can be used to effectively deal with this issue. Therefore, to enrich the theory of PFS, it is very necessary to investigate the fundamental properties of Pythagorean fuzzy information. In this paper, we first describe the change values of Pythagorean fuzzy numbers (PFNs), which are the basic components of PFSs, when considering them as variables. Then we divide all the change values into the eight regions by using the basic operations of PFNs. Finally, we develop several Pythagorean fuzzy functions and study their fundamental properties such as continuity, derivability, and differentiability in detail.

Hesitant fuzzy preference relation (HFPR) is an effective way to depict the decision makers’ preferences over the objects (alternatives or attributes) in the process of group decision making. Each component of the HFPR is characterized by several possible values and can express the decision makers’ hesitant information comprehensively. To make a decision with the HFPR, it is very necessary to find a proper technique for deriving the priority weights from the HFPR. In this paper, we use the error analysis as a tool to develop several straightforward methods for the priorities of the HFPR. We first define the expected value and the average value of each hesitant fuzzy element in the HFPR. Then based on the error analysis, we come up with the interval midpoint method, the average value method, and the difference method to derive the priority weights from the HFPR. After that, we discuss the relations among these methods, and utilize them and the possibility degree formula to develop an approach to decision making with the HFPR. Finally, we demonstrate the effectiveness and practicality of our approach through a case study concerning the investment problem in liquor enterprise.

Pythagorean fuzzy sets (PFSs), originally proposed by Yager, are a new tool to deal with vagueness with the square sum of the membership degree and the nonmembership degree equal to or less than 1, which have much stronger ability than Atanassov's intuitionistic fuzzy sets to model such uncertainty. In this paper, we modify the existing score function and accuracy function for Pythagorean fuzzy number to make it conform to PFSs. Associated with the given operational laws, we define some novel Pythagorean fuzzy weighted geometric/averaging operators for Pythagorean fuzzy information, which can neutrally treat the membership degree and the nonmembership degree, and investigate the relationships among these operators and those existing ones. At length, a practical example is provided to illustrate the developed operators and to make a comparative analysis.

Since hesitant fuzzy set was proposed, multi-attribute decision making (MADM) with hesitant fuzzy information, which is also called hesitant fuzzy MADM, has been a hot research topic in decision theory. This paper investigates a special kind of hesitant fuzzy MADM problems in which the decision data are expressed by several possible values, and the evaluative attributes are in different priority levels. Firstly, we introduce the definitions of hesitant fuzzy t-norm and t-conorm by extending the notions of t-norm and t-conorm to the hesitant fuzzy environment and explore their constructions by means of t-norms and t-conorms. Then motivated by the prioritized “or” operator (R. R. Yager, Prioritized aggregation operators, International Journal of Approximate Reasoning 2008;48:263–274), we develop the typical hesitant fuzzy prioritized “or” operator based on the developed hesitant fuzzy t-norms and t-conorms. In this operator, the degree of satisfaction of each alternative in each priority level is derived from a hesitant fuzzy t-conorm to preserve trade-offs among the attributes in the same priority level, and the priority weights of attributes are induced by a hesitant fuzzy t-norm to model the prioritization relationship among attributes. Furthermore, we apply the developed typical hesitant fuzzy prioritized “or” operator to solving the MADM problems in which the decision data are expressed by several possible values and the attributes are in different priority levels. In addition, two numerical examples are given to, respectively, illustrate the applicability and superiority of the developed aggregation operator by comparative analyses with previous research.

The innovative idea of Atanassov's intuitionistic fuzzy sets (A-IFSs) is to get a more comprehensive and detailed description of the ambiguity and uncertainty by introducing a membership function and a nonmembership function. Each element in an A-IFS is expressed by an ordered pair, which is called an intuitionistic fuzzy number (IFN). In this paper, we first describe the change values of IFNs when considering them as variables and classify these change values based on the basic operations for IFNs. Second, we depict the convergences of sequences of IFNs by the subtraction and division operations. Moreover, we develop some intuitionistic fuzzy functions (IFFs) and study in detail their continuities, derivatives, and differentials.

In this paper, we first review the existing entropy measures for hesitant fuzzy elements (HFEs) and demonstrate that the existing entropy measures for HFEs fail to effectively distinguish some apparently different HFEs in some cases. Then, we propose a new axiomatic framework of entropy measures for HFEs by taking fully into account two facets of uncertainty associated with an HFE (i.e., fuzziness and nonspecificity). We adopt a two-tuple entropy model to represent the two types of uncertainty associated with an HFE. Additionally, we discuss how to formulate each kind of uncertainty. For each of fuzziness and nonspecificity, some simple methods are provided to construct measures, which can well handle the problems in the existing entropy measures for HFEs. Several examples are given to illustrate each method, and comparisons with the existing entropy measures are also offered.

This paper mainly investigates a special kind of multicriteria decision–making problem, in which all the criteria can be divided into several hierarchies and the criteria in the higher hierarchy have priorities over those in the lower hierarchy. It implies that the loss of the higher priority criterion can't be compensated by the gain of the lower prioritized criteria. As we know, fuzzy measures can well represent the interactions between criteria. In this situation, we develop a new fuzzy measure called weakly ordered prioritized measure (WOPM) to express the priority rule among the weakly ordered prioritized criteria. On the basis of the WOPM, we use discrete Choquet integral to construct a new WOPM-guided aggregation (WOPMGA) operator. To understand the priority property of this aggregation operator deeply, we get all the criteria's Shapley values and make an analysis of all criteria's Shapley values with different parameter values. Through analysis, we can find that the WOPMGA operator has the properties of boundedness, idempotency and monotonicity. Finally, we give several practical examples to illustrate the effectiveness of this aggregation operator.

Intuitionistic fuzzy sets (IFSs) have attracted more and more scholars’ attention due to their powerfulness in expressing vagueness and uncertainty. In the course of decision making with IFSs, aggregation operators play a very important role since they can be used to synthesize multidimensional evaluation values represented as intuitionistic fuzzy values into collective values. This paper proposes a family of intuitionistic fuzzy hybrid weighted aggregation operators, such as the intuitionistic fuzzy hybrid weighted averaging operator, the intuitionistic fuzzy hybrid weighted geometric operator, the generalized intuitionistic fuzzy hybrid weighted averaging operator, and the generalized intuitionistic fuzzy hybrid weighted geometric operator. All these newly developed operators not only can weight both the arguments and their ordered positions simultaneously but also have some desirable properties, such as idempotency, boundedness, and monotonicity. To show the applications of our proposed intuitionistic fuzzy hybrid weighted aggregation operators, a simple schema for decision making with intuitionistic fuzzy information is developed. An example concerning the human resource management is given to illustrate the validity and applicability of the proposed method and also the hybrid weighted aggregation operators.

The necessity of dealing with uncertainty in real world problems has been a long-term research challenge that has originated different methodologies and theories. Fuzzy sets along with their extensions, such as type-2 fuzzy sets, interval-valued fuzzy sets, and Atanassov's intuitionistic fuzzy sets, have provided a wide range of tools that are able to deal with uncertainty in different types of problems. Recently, a new extension of fuzzy sets so-called hesitant fuzzy sets has been introduced to deal with hesitant situations, which were not well managed by the previous tools. Hesitant fuzzy sets have attracted very quickly the attention of many researchers that have proposed diverse extensions, several types of operators to compute with such types of information, and eventually some applications have been developed. Because of such a growth, this paper presents an overview on hesitant fuzzy sets with the aim of providing a clear perspective on the different concepts, tools and trends related to this extension of fuzzy sets.

Recently, a new model based on Pythagorean fuzzy set (PFS) has been presented to manage the uncertainty in real-world decision-making problems. PFS has much stronger ability than intuitionistic fuzzy set to model such uncertainty. In this paper, we define some novel operational laws of PFSs and discuss their desirable properties. For the multicriteria decision-making problems with PFSs, we propose an extended technique for order preference by similarity to ideal solution method to deal effectively with them. In this approach, we first propose a score function based comparison method to identify the Pythagorean fuzzy positive ideal solution and the Pythagorean fuzzy negative ideal solution. Then, we define a distance measure to calculate the distances between each alternative and the Pythagorean fuzzy positive ideal solution as well as the Pythagorean fuzzy negative ideal solution, respectively. Afterward, a revised closeness is introduced to identify the optimal alternative. At length, a practical example is given to illustrate the developed method and to make a comparative analysis.

After investigating prioritized multicriteria decision-making or aggregation problems in depth, we reveal that priority relationships among criteria may be incomplete. In a complex and practical multicriteria decision-making problem, the relationships of a pair of criteria can be multiform including priority relationships, except for independence. For such cases, we present a generalized prioritized multicriteria aggregation (GPMA) model with incomplete priority relationships in this paper. First, we elaborate GPMA problems, and describe them by using a dendriform structure after designing a top-to-bottom method to form priority hierarchies according to the given priority relationships. Furthermore, a common process for handling GPMA problems is proposed based on the premeditation that the weights associated with the lower-priority criteria need to be revised according to the satisfaction of the higher-priority criteria. At length, we develop two GPMA operators with respect to the GPMA with strictly and weakly ordered prioritizations, respectively, based on triangular norms, and we verify that both of them are aggregation functions.

Many aggregation operators involve sorting operations. People usually call them ordered aggregation operators, such as the ordered weighted averaging operators. The stability of these operators depends on the continuity of sorting operations. In this paper, we will show that the sorting operations have a very good continuity and thus the ordered aggregation operators are robust. However, the analysis of the continuity of sorting operations involving intuitionist fuzzy values is difficult. In this case, we may need to introduce a new topology and the results turn out to be rather complicated.

Ordered weighted averaging (OWA) operator's weights and orness measure play important roles in the application of the OWA operator to decision-making problems because the decision result may be different owing to the change in either of them. The aim of this paper is to investigate the influence that the change of OWA operator's weights or orness measure exerts on the decision result. We first give the range of the OWA operator's weights to keep the ranking order of alternatives or the optimal alternative unchanged. Then we make a sensitivity analysis to the orness measure to explore the dependency of the decision result on the orness measure. The results of analysis may provide a decision basis according to which decision makers are able to make a reasonable decision. Finally, a practical example is provided to illustrate the proposed sensitivity analysis methods.

Intuitionistic fuzzy set is a widely used tool to express the membership, nonmembership, and hesitancy information of an element to a set. To aggregate the intuitionistic fuzzy information, a lot of aggregation techniques have been developed, especially, the ones which reflect the correlations of the aggregated arguments are the hot research topics, among which Bonferroni mean (BM) is an important aggregation technique. However, the classical BM ignores some aggregation information and the weight vector of the aggregated arguments. In this paper, we introduce the generalized weighted BM and the generalized intuitionistic fuzzy weighted BM, both of which focus on the group opinion. Paying more attention to the individual opinions, we further define the generalized weighted Bonferroni geometric mean and the generalized intuitionistic fuzzy weighted Bonferroni geometric mean. Various families of the existing operators can be obtained when the parameters of the developed aggregation techniques are assigned different values. Finally, we propose an approach to multicriteria decision making on the basis of the proposed aggregation techniques and an example is also given to illustrate our results. © 2011 Wiley Periodicals, Inc.

We introduce the concepts of entropy and cross-entropy for hesitant fuzzy information, and discuss their desirable properties. Several measure formulas are further developed, and the relationships among the proposed entropy, cross-entropy, and similarity measures are analyzed, from which we can find that three measures are interchangeable under certain conditions. Then we develop two multiattribute decision-making methods in which the attribute values are given in the form of hesitant fuzzy sets reflecting humans' hesitant thinking comprehensively. In one method, the weight vector is determined by the hesitant fuzzy entropy measure, and the optimal alternative is obtained by comparing the hesitant fuzzy cross-entropies between the alternatives and the ideal solutions; in another method, the weight vector is derived from the maximizing deviation method and the optimal alternative is obtained by using the TOPSIS method. An actual example is provided to compare our methods with the existing ones. © 2012 Wiley Periodicals, Inc.

In this paper, we represent evaluation information by 2-dimension linguistic labels so as to avoid biased results and achieve high accuracy in multicriteria decision making. We analyze the relationship between a 2-dimension linguistic label and a common linguistic label, and then quantify a certain 2-dimension linguistic label by using a generalized triangular fuzzy number (TFN). On the basis of the mapping function from 2-dimension linguistic labels to the corresponding generalized TFNs and its inverse function, we develop a 2-dimension linguistic weighted averaging (2DLWA) operator and a 2-dimension linguistic ordered weighted averaging (2DLOWA) operator. Furthermore, we verify the feasibility of the 2DLWA and 2DLOWA operators by discussing their properties and investigating their applications to produce reliable decision results in multicriteria decision making under linguistic evaluations. Finally, an example of selecting the outstanding postgraduate dissertation(s) is used to illustrate the practicability and validity of these two 2-dimension linguistic aggregation techniques. © 2012 Wiley Periodicals, Inc.

Preference relations as simple and efficient information description tools have been widely used in practical decision-making problems. Intuitionistic preference relation, which is often met in real problems, is usually utilized to provide the experts' vague or fuzzy opinions over objects under uncertain circumstances. Owing to the limitations of the experts' professional knowledge and experience, the provided preferences in an intuitionistic preference relation are usually incomplete. Consequently, how to estimate the missing information in an expert's incomplete intuitionistic preference relation becomes a necessary step in a decision-making process. In this paper, we define the concept of multiplicative consistent intuitionistic preference relation and develop two estimation algorithms. The first algorithm is used to estimate the missing elements using only the known preference values in an acceptable incomplete intuitionistic fuzzy preference relation with the least judgments. The second one is given for the estimation of missing elements of the acceptable incomplete intuitionistic fuzzy preference relations with more known judgments. The advantages of the developed algorithms over the existing one are detailedly analyzed, and some examples are provided to illustrate the solution processes of the algorithms and to verify their practicality and superiority. © 2011 Wiley Periodicals, Inc.

A hesitant fuzzy set, allowing the membership of an element to be a set of several possible values, is very useful to express people's hesitancy in daily life. In this paper, we define the distance and correlation measures for hesitant fuzzy information and then discuss their properties in detail. These measures are all defined under the assumption that the values in all hesitant fuzzy elements (the fundamental units of hesitant fuzzy sets) are arranged in an increasing order and two hesitant fuzzy elements have the same length when we compare them. We can find that the results, by using the developed distance measures, are the smallest ones among those when the values in two hesitant fuzzy elements are arranged in any permutations. In addition, the derived correlation coefficients are based on different linear relationships and may have different results. © 2011 Wiley Periodicals, Inc.

In practical decision making problems, whether an alternative (strategy) of a decision maker is good or not relies on not only influences of its natural characters and external environment but also interactions with the decision maker's rivals. Thus for the sake of more accurate and actual assessments, in this paper we analyze alternatives (strategies) quantitatively based on both multi-attribute decision making methods and game theory. A multi-attribute aggregation based game model is then constructed. For general games, we first divide strategies into four forms after investigating the traits of practical game problems, and analyze relationships of these forms of strategies. Then four classifications of general game problems are presented according to the differences of strategies involved. Based on the game model, general methods to solve these four classifications of games are put forward, and relevant examples are taken to illustrate the flexibility of the methods. Moreover, we investigate the methods for solving multicriteria games, and finally extend them to general forms on the basis of the game model. © 2011 Wiley Periodicals, Inc.

We first develop a series of intuitionistic fuzzy point operators, and then based on the idea of generalized aggregation (Yager RR. Generalized OWA aggregation operators. Fuzzy Optim Decis Making 2004;3:93–107 and Zhao H, Xu ZS, Ni MF, Liu SS. Generalized aggregation operators for intuitionistic fuzzy sets. Int J Intell Syst 2010;25:1–30), we develop various generalized intuitionistic fuzzy point aggregation operators, such as the generalized intuitionistic fuzzy point weighted averaging (GIFPWA) operators, generalized intuitionistic fuzzy point ordered weighted averaging (GIFPOWA) operators, and generalized intuitionistic fuzzy point hybrid averaging (GIFPHA) operators, which can control the certainty degrees of the aggregated arguments with some parameters. Furthermore, we study the properties and special cases of our operators. © 2010 Wiley Periodicals, Inc.

Intuitionistic fuzzy numbers are very useful for experts to depict in depth their fuzzy preference information over objects. In this work, we investigate multiple attribute group decision-making problems in which the attribute values provided by experts are expressed in intuitionistic fuzzy numbers, each of which is composed of a membership degree, a nonmembership degree and a hesitancy degree, and the weight information about both the experts and the attributes is to be determined. We first make different types of attribute values uniform so as to facilitate interattribute comparisons and employ the simple additive weighting method to fuse all the individual opinions into the group one. We then develop two nonlinear optimization models, one minimizing the divergence between each individual opinion and the group one, and the other minimizing the divergence among the individual opinions, from which two exact formulae can be obtained to derive the weights of experts. Similarly, from the viewpoint of maximizing group consensus, we establish a nonlinear optimization model based on all the individual intuitionistic fuzzy decision matrices to determine the weights of attributes. The simple additive weighting method is used to aggregate all the intuitionistic fuzzy attribute values corresponding to each alternative, and then the score function and the accuracy function are employed to rank and select the given alternatives. Moreover, we extend all the above results to interval intuitionistic fuzzy situations, and finally apply the developed models to an air-condition system selection problem. © 2010 Wiley Periodicals, Inc.

The generalized ordered weighted averaging (GOWA) operators are a new class of operators, which were introduced by Yager (Fuzzy Optim Decision Making 2004;3:93–107). However, it seems that there is no investigation on these aggregation operators to deal with intuitionistic fuzzy or interval-valued intuitionistic fuzzy information. In this paper, we first develop some new generalized aggregation operators, such as generalized intuitionistic fuzzy weighted averaging operator, generalized intuitionistic fuzzy ordered weighted averaging operator, generalized intuitionistic fuzzy hybrid averaging operator, generalized interval-valued intuitionistic fuzzy weighted averaging operator, generalized interval-valued intuitionistic fuzzy ordered weighted averaging operator, generalized interval-valued intuitionistic fuzzy hybrid average operator, which extend the GOWA operators to accommodate the environment in which the given arguments are both intuitionistic fuzzy sets that are characterized by a membership function and a nonmembership function, and interval-valued intuitionistic fuzzy sets, whose fundamental characteristic is that the values of its membership function and nonmembership function are intervals rather than exact numbers, and study their properties. Then, we apply them to multiple attribute decision making with intuitionistic fuzzy or interval-valued intuitionistic fuzzy information. © 2009 Wiley Periodicals, Inc.

Harmonic mean is a conservative average, which is widely used to aggregate central tendency data. In the existing literature, the harmonic mean is generally considered as a fusion technique of numerical data information. In this paper, we investigate the situations in which the input data are expressed in fuzzy values and develop some fuzzy harmonic mean operators, such as fuzzy weighted harmonic mean operator, fuzzy ordered weighted harmonic mean operator, fuzzy hybrid harmonic mean operator, and so on. Especially, all these operators can be reduced to aggregate interval or real numbers. Then based on the developed operators, we present an approach to multiple attribute group decision making and illustrate it with a practical example. © 2008 Wiley Periodicals, Inc.

Yager [IEEE Trans Syst Man Cybern B 2004;34:1952–1963] introduced a continuous interval argument OWA (C-OWA) operator, which extends the ordered weighted averaging (OWA) operator, introduced by Yager [IEEE Trans Syst Man Cybern B 1988;18:183–190], to the case in which the given argument is a continuous valued interval rather than a finite set of values. In this article, we utilize the C-OWA operator to derive the priority vector of an interval fuzzy preference relation and then develop a practical approach to solving the decision-making problem with interval fuzzy preference relation. Finally, a numerical example is provided to demonstrate the practicability and efficiency of the developed approach. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 1289–1298, 2006.

The ordered weighted aggregation (OWA) operator has received more and more attention since its appearance. One key point in the OWA operator is to determine its associated weights. In this article, I first briefly review existing main methods for determining the weights associated with the OWA operator, and then, motivated by the idea of normal distribution, I develop a novel practical method for obtaining the OWA weights, which is distinctly different from the existing ones. The method can relieve the influence of unfair arguments on the decision results by weighting these arguments with small values. Some of its desirable properties have also been investigated. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 843–865, 2005.

In this work, we first make a survey of the existing main aggregation operators and then propose some new aggregation operators such as the induced ordered weighted geometric averaging (IOWGA) operator, generalized induced ordered weighted averaging (GIOWA) operator, hybrid weighted averaging (HWA) operator, etc., and study their desirable properties. Finally, we briefly classify all of these aggregation operators. © 2003 Wiley Periodicals, Inc.

The ordered weighted averaging (OWA) operator was introduced by Yager1 to provide a method for aggregating several inputs that lie between the max and min operators. In this article, we investigate the uncertain OWA operator in which the associated weighting parameters cannot be specified, but value ranges can be obtained and each input argument is given in the form of an interval of numerical values. The problem of ranking a set of interval numbers and obtaining the weights associated with the uncertain OWA operator is studied. © 2002 Wiley Periodicals, Inc.

The ordered weighted averaging (OWA) operator was introduced by Yager.1 The fundamental aspect of the OWA operator is a reordering step in which the input arguments are rearranged in descending order. In this article, we propose two new classes of aggregation operators called ordered weighted geometric averaging (OWGA) operators and study some desired properties of these operators. Some methods for obtaining the associated weighting parameters are discussed, and the relationship between the OWA and DOWGA operators is also investigated. © 2002 Wiley Periodicals, Inc.