Individual interactions and conformity play a crucial role in shaping group opinions and influencing the decision-making process. This article introduces a novel opinion evolution model to simulate the impact of conformity on group opinion formation, focusing on weight allocation, relationship propagation, and evolution analysis. In the weight allocation phase, individual weights are evaluated using network structure and the PageRank algorithm. For relationship propagation, indirect trust relationships are computed via a weighted average method, leading to a more precise social network model. In the evolution analysis, an improved Hegselmann–Krause (HK) model demonstrates evolutionary stability. Two types of conformity behavior are simulated: active conformity behavior within clusters and passive conformity behavior via opinion leaders across clusters. Experimental studies on public health policy validate the effectiveness of the proposed model. The results show that this model more accurately captures the complex behavioral patterns of individuals in large-scale social networks and exhibits strong evolutionary stability.

One of the most significant goals that must be guaranteed for high-speed railway systems is safety. A new model is constructed for high-speed railway accident analysis with interval type-2 fuzzy DEMATEL method based on the 24Model. Specifically, the 24Model is introduced to identify the accident factors from four stages at the individual and organizational levels. The DEMATEL method is combined with interval type-2 fuzzy sets to analyze the key causations of the accident considering the interrelationships between factors under an uncertain environment. A specific case of the “7.23” Yong-Tai-Wen railway accident is employed to validate the proposed model. The analysis results show that poor management of the ministry of railways, imperfect rules and standards and little supervision or inspection are core organizational factors that lead to accidents. Besides, unsafe material conditions and unsafe human conditions are the direct causes of the accident which are easily influenced by other accident factors. The relevant managers need to strengthen the inspection to ensure the safe operation of the high-speed railway.

Cross-efficiency evaluation with the data envelopment analysis (DEA) model is an effective way to assess performance and provide a complete ranking of decision-making units. However, it is generally assumed that decision makers are perfectly rational in the cross-efficiency model, which fails to consider the subjective preferences of decision-makers. Moreover, the arithmetic average method is usually adopted to aggregate efficiency scores in traditional cross-efficiency methods, which underestimates the importance of self-evaluation. To address these issues, we extend cross-efficiency with the DEA model by incorporating prospect theory and the distance entropy function. First, we calculate the prospect values of decision-making units to describe the non-rational subjective preferences under the risk of decision-makers. Second, based on prospect cross-efficiency, a new distance entropy function is developed to aggregate the ultimate prospect cross-efficiency values. More specifically, some traditional cross-efficiency evaluation models can be considered special cases of prospect cross-efficiency models with appropriate adjustments to the parameters. Finally, an empirical example is used to evaluate the prospect cross-efficiency results with the high-tech industries of 29 provinces in China to illustrate the applicability and effectiveness of the proposed model in ranking observations.

Social network group decision-making (SNGDM) develops rapidly because of the popularity of online connections among experts. The current SNGDM studies mainly focus on the influence of trust on the evolution of consensus opinions through a complex process. Some situations are ignored in the existing research when trust may negatively affect the decision-making process and trust relationships may be developed or interrupted during consensus negotiation processes. To overcome such limitations, we propose a consensus model based on the trust-consensus multiplex network by combing trust social network analysis and consensus evolution networks (CENs). We design an interaction mechanism between trust and consensus based on the dynamic experts’ influence which is computed by the multiplex PageRank centrality measure, especially focusing on the negative impact of trust on consensus. Besides, we compute consensus levels based on the density and intensity of CENs to determine when should the negotiation end. The proposed model can not only consider the mutual influence between trust and consensus in SNGDM, but also detect and analyze the negative influence of trust on consensus. An example examines the effectiveness of the proposed model and a comparative analysis shows its flexibility for studying the complex consensus situation of SNGDM.

In large-scale group decision making (LSGDM), appropriate clustering analysis is important to consensus reaching since it can reduce the interactive complexity among individuals. According to the traditional clustering method, a conflict may arise between the consensus reaching levels and total adjustment costs within clusters when individuals have different unit adjustment cost, which reflects their willingness to make concessions. Since this conflict may aggravate the consensus complexity, we propose a new$K$-means clustering method that considers both preferences and the preference adjustment cost. The preference adjustment cost is attached to preferences with a parameter that can be determined by balancing this conflict. Because of such conflict, the proposed clustering algorithm can improve the similarity of intracluster individuals on the preference adjustment cost by offsetting some acceptable consensus reaching levels within clusters. According to the proposed clustering algorithm, individuals who have both similar preferences and adjustment willingness are classified into the same clusters. In this way, the moderator can provide similar compensation strategies for intracluster individuals, which will decrease the adjustment complexity. A practical case study of team construction examines the application of the proposed algorithm, and the related comparative analysis shows that it is convenient for managers to persuade individuals to reach a consensus under the improved clustering results.

In most existing linguistic distribution multiple criteria group decision-making (MCGDM) models, the sample size of group linguistic distribution assessment (LDA) is often neglected, and the decision makers are viewed to be independent. The probabilistic distribution is closely related to the sample size, and there exist more or fewer interference effects among different individual opinions. Thus, this article develops a quantum framework for modeling interference effects in the linguistic distribution MCGDM process. First, considering the sample size information, we redefine the LDAs and provide a new computational model for them. Second, to integrate a series of LDAs, the linguistic distribution weighted averaging (LDWA) operator is presented. Mathematic proofs reveal that the new LDWA operator can not only ensure the information integrality but also can avoid the conjunctive and disjunctive operations. Third, to explore the interference effects among individual opinions, a quantum framework is constructed. In this process, individual opinions are viewed as various wave functions occurring synchronously. They interfere with each and influence the aggregation result. Finally, an illustrative example of Internet finance soft power evaluation is provided to verify the effectiveness. Sensitivity and comparative analyses are also implemented to assess the stability and validity of our method.

Large-scale group decision-making (LSGDM) solution is usually based on the clustering analysis process (CAP) and consensus reaching process (CRP). However, CAP and CRP can be contradictory since CAP is performed based on the differences between potentially small groups and CRP is conducted to improve the overall similarity of a large group. To balance CAP and CRP, a dynamic clustering analysis process (DCAP) based on consensus evolution networks is proposed. A clustering algorithm proposed based on community detection method can be used to handle the diverse network structures with dynamic consensus thresholds. The clustering validity based on the intracluster consensus levels in subgroups and the intercluster consensus level among subgroups is evaluated. Then, the DCAP after each feedback adjustment round in CRP is reanalyzed. In such a way, effective clustering can also be found after a satisfying consensus is reached. Finally, a case study shows the availability of this approach and comparative analyses are provided to highlight the advantages from both theoretical and numerical perspectives.

The social network group decision-making is popular due to the advantages of social relationships in the consensus reaching process, especially the trust relationships. To explore the effects of trust on consensus, some minimum cost consensus models are proposed based on implicit trust between individuals and the moderator. The implicit trust is computed based on the similarity of opinions and it is implied into the traditional minimum cost consensus model to obtain a new quadratic programming problem and the related dual problem. The weights of individuals can be determined based on implicit trust and can be used to modify the possible deviations among individuals’ adjustment costs. A numerical example and the comparative analysis are given to analyze the effectiveness of the proposed models, which suggests that individuals are willing to give up some benefit to reach consensus due to their implicit trust to the moderator and make minor revisions to their adjustment costs due to their implicit trust to each other.

Enhancing the reliability of machine tools and preventing the potential failures in the manufacturing process is one of the most important tasks for the development of industry. The failure mode and effects analysis (FMEA) is the well-known and widely utilized approach to identify and evaluate potential failures for preventing risk in various enterprises. It is a group-oriented technique ordinarily conducted by a group of experts from related fields. Obviously, an effective approach should be developed, which is used to integrate risk evaluation information from multiexperts by considering group and individual risk attitudes. This article proposes a hybrid FMEA framework for addressing machine tool risk analysis problem by integrating cloud model, Choquet integral, and gained and lost dominance score (GLDS) method. In this framework, an improved Shapley cloud-Choquet weighting averaging operator is defined to fuse random and uncertain risk information by considering the correlations among experts. An extended GLDS method with developed Choquet integral based on distance measure of normal clouds is presented to prioritize the risk priority of each failures, in which the group and individual risk attitude and risk interactions are considered simultaneously. Finally, a real risk analysis of machine tool in a machine tool industry is introduced to illustrate the application and feasibility of the proposed approach, and comparison and sensitivity studies are also conducted to validate the effectiveness of the hybrid framework.

Abstract Trust is crucial for purchasing decisions in social commerce. However, inexperienced users may not have a direct trust relation to experienced users in practice. Besides, users tend to give their trust degrees to others with linguistic labels rather than crisp values. To evaluate the trust degree for inexperienced users to experienced ones, we propose an interval type‐2 fuzzy trust evaluation model in this paper. Interval type‐2 fuzzy linguistic labels are used to represent trust degree among users. An interval type‐2 fuzzy Algebraic t‐norm is addressed to compute propagative trust degrees. Considering the effect of trust path length, the induced ordered weighted averaging (IOWA) operator is extended to aggregate the interval type‐2 fuzzy trust degrees obtained from multiple paths. In addition, the final interval type‐2 fuzzy trust degree is transferred into the corresponding linguistic label to help users make decisions more naturally. Finally, a case study in social commerce and a related comparison are given to verify the effectiveness of the proposed model.

The theory of fuzzy sets considers that everything exhibits some elasticity and is a matter of degree. When fuzzy numbers are used to evaluate the judgements of decision makers (DMs) in pairwise comparisons of alternatives following the analytic hierarchy process, the flexibility experienced by DMs has been exhibited. In order to capture this aspect of flexibility, it is important to know how to realize the flexibility degree of fuzzy numbers and further present a method of realizing its quantification. In this paper, a definition of the flexibility degree of fuzzy numbers is proposed. Some formulas are proposed to quantify the flexibility and rigidity degrees of interval numbers, triangular fuzzy numbers, and trapezoidal fuzzy numbers. A group decision making (GDM) model is developed under the consideration of the flexibility of DMs. By considering the effects of the applied scale and the reciprocal relation, the flexibility degree of interval multiplicative reciprocal comparison matrices is further defined, which is used to evaluate the flexibility degree of the DM involved in the decision process. An RD-IOWGA operator is proposed to aggregate individual interval multiplicative reciprocal matrices by associating more importance to that with less flexibility. A new algorithm is shown to solve GDM problems with interval multiplicative reciprocal preference relations. Numerical studies are carried out to illustrate the new definitions and offer some comparative analysis. The observations reveal that the developed consensus method can be used to model the GDM with a dominant position.

In this paper, a closed form type reduction method for piecewise linear interval type-2 fuzzy sets and root finding methods are applied in centroid type-reduction for piecewise linear general type-2 fuzzy sets. When the lower and upper bounds of FOUs of the general type-2 fuzzy sets are provided, the centroid computation methods in this paper is accurate and computational efficient, and is represented by a closed-form formula, which provide a way to perform theoretical analysis for the general type-2 fuzzy sets.

This paper presents an improved algorithm for multistage portfolio optimization model in fuzzy environment. Firstly, the basic multistage portfolio selection problem is introduced by using multiple criteria, including investment return, transaction cost and risk. Then, a TOPSIS compromised programming (TOPSCP) approach is designed to transform the proposed model into single objective model. Finally, a genetic algorithm is devised for obtaining optimal solutions. Furthermore, a numerical example is given to illustrate the advantages of the algorithm.

In this paper, we introduce the concept called fuzzy non-preferred relation where the elements are interpreted as the non-preferred intensity of one alternative over another one. We divide an intuitionistic preference relation into the fuzzy preference relation part and the fuzzy non-preferred relation part . Based on this division, we propose a new definition of additive consistency of intuitionistic preference relation based on the traditional one. Then, we construct an optimization model for deriving intuitionistic fuzzy weights which can be ranked easily in individual decision making environment. And we develop an optimization model for deriving the collective intuitionistic fuzzy weights to address group decision making problems. Finally, two numerical examples are provided to illustrate the developed approaches.

Three-way decision method is a kind of new developed uncertain decision theory, in which a set of objects can be divided into three regions, called the acceptance, rejection and uncertainty regions, respectively. However, seldom researches that involve the fusion of linguistic information and three-way decisions are found. In this paper, we propose a novel decision method, called linguistic three-way decisions, in which the alternatives are rated in linguistic forms and divided into acceptance, rejection and uncertainty regions. The alternatives in the uncertainty region are especially discussed. The linguistic terms are transformed into fuzzy numbers and then aggregated. Finally, a linguistic recommendation example is provided to illustrate the practicality and validity of the proposed methods.

In this paper, we investigate the multiple attribute group decision making (MAGDM) problems based on grey system theory and utility theory to accommodate situations where attribute values take the form of interval type-2 fuzzy sets (IT2FSs). Motivated by the idea of grey incidence analysis theory, we propose two new grey incidence degrees for aggregating the uncertain information and further extend the concept to accommodate interval type-2 fuzzy environment. Meanwhile, based on the flexible three parameter (FTP) utility function, we develop an interval type-2 fuzzy ranking weighted utility averaging (IT2FRWUA) operator, some desirable mathematical properties are discussed in detail. Then, we further integrate the proposed grey incidence degrees and the IT2FRWUA operator to develop two methods for handling MAGDM problems within the context of IT2FSs. Finally, some practical examples are provided to illustrate the practicality and effectiveness of the proposed methods.

In order to overcome the drawbacks of AHP in solving complex decision-making problems, ANP is applied in the evaluation of enterprise technology innovation ability. Due to Interval type-2 fuzzy sets can handle uncertainty of linguistic variables in a more flexible and precisely way than Type-1 fuzzy sets with their fuzzy membership functions, a fuzzy ANP method with interval type-2 fuzzy sets is proposed to evaluate the enterprise technology innovation ability. A new ranking method based on the centroid is applied in processing relationships between interval type-2 fuzzy sets. And the proposed method is used in the numerical examples and ranking results are obtained.

Many group decision making (GDM) problems under uncertain environments have vague and imprecise information in linguistic variable formats. Multi-granularity linguistic method with the process of symbolic computation is a typical tool to solve such problems, and can be associated with the popular computing with words domain. In the paper, we present numerical two-scale model which is the extension of symbolic computation model. Firstly, numerical two-scale representation model is proposed with two scale measurements. One is used to reflect the order of linguistic variable and the other is used to model vagueness. Secondly, we give numerical two-scale computational model in group decision making. We discuss the rules of symbolic method which directly compute with the two numerical measurements. Finally, some aggregation operators are developed. The new model has the advantage of avoiding the vague information losing without adding the calculation difficulty.

In this paper, we investigate the multiple attribute decision making problems where the decision-making information and attribute weight vector are both given by the interval-valued intuitionistic fuzzy number (IVIFN). We introduce a mathematical model to obtain the comprehensive value of each alternative by the form of IVIFN. Then we utilize the TOPSIS method to rank all the alternatives. Finally, an illustrative example is used to illustrate applicability of the proposed method.

The linguistic weighted average (LWA) of interval type-2 fuzzy sets is an extension of the fuzzy weighted average (FWA) of type-1 fuzzy sets. Currently, the commonly used methods of both FWA and LWA are based on a-cuts decomposition of the type-1 of type-2 fuzzy sets, which involves large amount of calculations, and the result is not accurate. In this paper, we propose a new algorithm to obtain the analytical solution methods, which is more accurate and efficient than the current a-cuts methods. Some properties of the algorithms are discussed. A numerical example is used to illustrate our new proposed algorithms.

In this paper, we present the interval type-2 fuzzy rational degree to measure the similarity of the interval type-2 fuzzy sets and apply it to multiple attribute decision making with interval type-2 fuzzy information. First, we introduce the concept of the interval type-2 fuzzy metric spaces, which include interval type-2 fuzzy distance space and interval type-2 fuzzy inner product space, respectively. Based on which, we derive the distance measure and inner product of interval type-2 fuzzy sets, respectively. Then, we introduce the axiomatic definition of the interval type-2 fuzzy rational degree and propose an interval type-2 fuzzy rational degree formula. Moreover, we construct the mathematical optimal model based on two types of interval type-2 fuzzy measures (center of gravity and fuzziness) to determine the optimal attribute weights. Based on the interval type-2 fuzzy rational degree and the optimal weights solution model we proposed, an approach to multiple attribute decision making under interval type-2 fuzzy environment is developed. Finally, an illustrative example is given to demonstrate the practicality and effectiveness of our method.

Although there are many methods for ranking type-1 fuzzy numbers, most of which exist some limitations. In this paper, we firstly propose a new distance measure based method to rank type-1 fuzzy numbers, which defines two formats of possibility mean and variation coefficient. It not only clearly discriminates the ranking of the type-1 fuzzy numbers especially for the reasonable ranking of symmetric type-1 fuzzy numbers, but also satisfies the consistence of the ranking with their images. Then, we extend such ranking method to interval type-2 fuzzy numbers. For ranking both the type-1 fuzzy numbers and interval type-2 fuzzy numbers, the proposed methods are easy to understand and their computations are simple. Several typical examples are used to illustrate our new ranking methods for type-1 fuzzy numbers and interval type-2 fuzzy numbers.

Interval-valued intuitionistic fuzzy preference relation is a useful tool to express decision maker's interval-valued intuitionistic fuzzy preference information over criteria in the process of multi-criteria decision making. How to derive the priority weights from an interval-valued intuitionistic fuzzy preference relation is an interesting and important issue in decision making with interval-valued intuitionistic fuzzy preference relation(s). In this paper, some new concepts such as interval-valued interval fuzzy sets, interval-valued interval fuzzy preference relation and consistent interval-valued intuitionistic fuzzy preference relation, are defined, and the equivalent interval-valued interval fuzzy preference relation of interval-valued intuitionistic fuzzy preference relation is given. Then a method for estimating criteria weights from interval- valued intuitionistic fuzzy preference relations is developed, two numerical examples are provided to illustrate the developed method.

We propose a parametric weighting function for weighted ordered weighted averaging (WOWA) operator in decision making under uncertainty and risk environment. The weighting function is not only related to the attitude of the decision makers, but also to the event probability. Under the proposed method, the decision makers underweight small probability gains and overweight large probability gains, which is right in consistence with the psychology of the decision makers in the prospect theory. Meanwhile, it generalizes two specific weighting functions and proposed by Torra (International Journal of Intelligent Systems 12 (1997) 153–166). Moreover, the expected utility becomes a special case of the parametric weighting function for WOWA operator. To illustrate its application, an example of introducing and pricing new product is given, which shows the optimal strategies corresponding to different parameter values r.

Karnik-Mendel (KM) algorithms are important tools for the centroid computation or type reduction in the type-2 fuzzy logic. To release the possible computational bottleneck of the iteration of KM algorithms, some uncertainty bound methods are proposed. In the present paper, the current uncertainty bounds are extended and improved with three new uncertainty bound methods for the centroid computation of interval type-2 fuzzy sets. The New uncertainty bound methods can be further improved if some prior information is available or be be improved from the current bound values. Theoretical analysis proves the validity of our new approaches.

We analyze the application of regular increasing monotonic(RIM) quantifiers for parametric weighted ordered weighted average (WOWA) operator with piecewise linear function in decision problems. It can be used to represent various attitudes of the decision makers in forms of risk pessimistic, risk neutral and risk optimistic, respectively. Then, some properties of parametric WOWA operator with piecewise linear function are provided. Expected utility theory becomes a special case of parametric WOWA operator with risk neutral attitude. Different attitudes of the decision makers can be implemented by changing the values of parameter in parametric WOWA operators according to application context. Finally, the paper illustrates our theoretical results with an example of introducing and pricing new product.

Based on the current developments of fuzzy weighted average and generalized fuzzy weighted average, especially Karnik-Mendel (KM) algorithms and the works of Wang and Luo (Generalised fuzzy weighted mean and its applications, International Journal of General Systems, 2009, 38, 533–546), the paper analyses some properties of the generalized fuzzy weighted mean (GFWM) and proposes analytical solution method for the GFWM problem. The solution processes and the final solution forms are more simple and accurate than the current GFWM solution, which use samples of the alpha-cuts and get the final solutions approximately. Numerical examples illustrate the analytical solution computing processes for fuzzy weighted average (FWA), fuzzy weighted harmonic mean (FWHM), fuzzy weighted geometric mean (FWGM) and GFWM with exponential and power functions, respectively.

The most preferred OWA (MP-OWA) operator is a new method to aggregate preference information with crisp numbers, whose weights are related with the frequency of the most preferred assessment to each criteria. However, people are usually not able to estimate their preference degrees with crisp number, since they have a vague knowledge about the preference assessment. In this paper, we propose a 2-tuple linguistic MP-OWA (LMP-OWA) operator. It is useful because it can be used to make decision with linguistic preference relations, and the weighting vector is not only connected to the maximum frequency of the assessment to the criteria, but also to the assessment values. Meanwhile, we introduce the parameterized 2-tuple LMP-OWA operator and the parameterized 2-tuple LMP-OWA operator with power function, which provide multiple aggregation results for decision makers to select. The paper ends up with an example of decision making with linguistic preference relations in movie recommender system.

We define some operations over Atanassov's intuitionistic fuzzy sets (AIFSs), such as Einstein intersection, Einstein product, Einstein scalar multiplication and Einstein exponentiation, etc., and then define new concentration and dilation of AIFSs. These definitions will be useful while dealing with various linguistic hedges like “very”, “more or less”, “highly”, “very very” etc., involved in the problems under intuitionistic fuzzy environment. We also prove some propositions and present some examples in this context.

Type reduction (TR) followed by defuzzification is commonly used in interval type-2 fuzzy logic systems (IT2 FLSs). Because of the iterative nature of TR, it may be a computational bottleneck for the real-time applications of an IT2 FLS. This has led to many direct approaches to defuzzification that bypass TR, the simplest of which is the Nie-Tan direct defuzzification method (NT method). This paper provides some theoretical analyses of the NT method that answer the question “Why is the NT method good to use?” This paper also provides a direct relationship between TR followed by defuzzification (using KM algorithms) and the NT method. It also provides an improved NT method. Numerical examples illustrate our theoretical results and suggest that the NT method is a very good way to simplify an interval type-2 fuzzy set.

In this paper, we give two direct centroid computation methods for fuzzy numbers: One is to use the membership function and the other is to use the alpha-cuts. Compared with the current centroid computation methods, the new methods are simple both in expression and computation. Weighted samples computation method are also proposed to improve the computational accuracy with numerical integration technique. Three examples illustrate the application of proposed methods.

Karnik-Mendel (KM) algorithms type reduction are commonly used in the centroid computation of interval type-2 fuzzy logic systems (IT2 FLSs). Some properties and improvements of the KM algorithms have be proposed. This paper proposed a new iteration formula for the centroid computation of interval type-2 fuzzy sets, which is faster than the current computation methods and can be used as a direct method for the centroid computation of interval type-2 fuzzy sets.

Karnik-Mendel (KM) algorithms used for type reduction (TR) followed by defuzzification are commonly used in interval type-2 fuzzy logic systems (IT2 FLSs). Up until now no closed-form solution has existed for KM+defuzzification. Our paper gives, for the first time, a Taylor-series approximation to KM+defuzzification. The first term in the approximation is the Nie-Tan (NT) formula, and the second-term (which is actually the third-order term in the Taylor series) is a correction to it. One very interesting fact is that the Taylor series approximation is itself iterative (as is the KM Algorithm) in that the second term depends on the first term. Simulations show that the first term is a very good approximation to KM+defuzzification.

Aggregation of fuzzy information is a new branch of Atanassov's intuitionistic fuzzy set (AIFS) theory, which has attracted significant interest from researchers in recent years. In this paper, we treat the intuitionistic fuzzy aggregation operators with the help of Einstein operations. We first introduce some new operations of AIFSs, such as Einstein sum, Einstein product, and Einstein scalar multiplication. Then, we develop some intuitionistic fuzzy aggregation operators, such as the intuitionistic fuzzy Einstein weighted averaging operator and the intuitionistic fuzzy Einstein ordered weighted averaging operator, which extend the weighted averaging operator and the ordered weighted averaging operator to aggregate Atanassov's intuitionistic fuzzy values, respectively. We further establish various properties of these operators and analyze the relations between these operators and the existing intuitionistic fuzzy aggregation operators. Moreover, we give some numerical examples to illustrate the developed aggregation operators. Finally, we apply the intuitionistic fuzzy Einstein weighted averaging operator to multiple attribute decision making with intuitionistic fuzzy information.

The ordered weighted averaging (OWA) determination method with stress function was proposed by Yager, and it makes the OWA operator elements scatter in the shape of the stress function. In this paper, we extend the OWA determination with the stress function method using an optimization model. The proposed method transforms the OWA optimal solution elements into the interpolation points of the stress function. The proposed method extends the basic form of the stress function method with both scale and vertical shift transformations. We also explore a number of properties of this optimization-based stress function method. The OWA operator optimal solution elements can distribute as the shape of the given stress function in a parameterized way, in which case, the solution always possesses the arithmetic average operator as a special case.

The supplier selection is one of the most important components of production and logistics management for many companies. The supplier selection problem involves different kinds of words in which all of the criteria are weighted using one kind of words and the performance evaluations for all sub-criteria are another kind of words. How to aggregate these words without losing information is a very daunting task using a type-1 fuzzy set (Tl FS) approach. This paper applies a new methodology- Perceptual Computer (Per-C)-to help solve this hierarchical group multi-criteria decision making problem. The Per-C has three components: encoder, computing with words (CWW) engine and decoder. First, the interval approach (IA) is used to obtain interval type-2 fuzzy set (IT2 FS) word models for the words in a pre specified vocabulary. Second, a linguistic weighted average (LWA) is used to aggregate all the words modeled by IT2 FSs. Finally, a centroid-based ranking method is used to rank the supplier selections, and a similarity measure is used to obtain similarities of the supplier selections. A committee of judges organized with P decision makers and K experts decides the winning supplier selection as the one with the highest ranking and least similarity to other suppliers.

This paper provides root finding methods for the centroid computation of interval type-2 fuzzy sets and general type-2 fuzzy sets with linear secondary membership function. When the FOUs of the type-2 fuzzy sets are provided, the centroid computation methods in this paper is accurate and computational efficient, which provide new methods for the type-2 fuzzy systems computation.

Interval type-2 fuzzy sets, each of which is characterized by the footprint of uncertainty, are a very useful means to depict the decision information in the process of decision making. In this article, we investigate the decision making problems in which all the information provided by the decision maker is expressed as interval type-2 fuzzy decision matrix where each of the elements is characterized by interval type-2 fuzzy set, and the information about attribute weights is partially known, which may be constructed by various forms. We first utilize the average centroid measure to calculate the average centroid of each attribute value and construct the average centroid matrix of the interval type-2 fuzzy decision matrix. Based on the average centroid matrix and the given attribute weight information, we establish some optimization models to determine the weights of attributes. Furthermore, we develop a procedure to identify the best alternative(s). Finally, we give an illustrative example.

Intuitionistic fuzzy information aggregation plays an important part in Atanassov's intuitionistic fuzzy set theory, which has emerged to be a new research direction receiving more and more attention in recent years. In this paper, we first introduce some operations on intuitionistic fuzzy sets, such as Einstein sum, Einstein product, Einstein exponentiation, etc., and further develop some new geometric aggregation operators, such as the intuitionistic fuzzy Einstein weighted geometric operator and the intuitionistic fuzzy Einstein ordered weighted geometric operator, which extend the weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator to accommodate the environment in which the given arguments are intuitionistic fuzzy values. We also establish some desirable properties of these operators, such as commutativity, idempotency and monotonicity, and give some numerical examples to illustrate the developed aggregation operators. In addition, we compare the proposed operators with the existing intuitionistic fuzzy geometric operators and get the corresponding relations. Finally, we apply the intuitionistic fuzzy Einstein weighted geometric operator to deal with multiple attribute decision making under intuitionistic fuzzy environments. © 2011 Wiley Periodicals, Inc.

Based on a new continuous Karnik-Mendel (KM) algorithm expression, this paper proves that the centroid computation of an interval type-2 fuzzy set using KM algorithms is equivalent to the Newton-Raphson method in root-finding, which reveals the mechanisms in KM algorithm computation. The theoretical results of KM algorithms are re-obtained. Different from current KM algorithms, centroid computation methods that use different root-finding routines are provided. Such centroid computation methods can obtain the exact solution and are different from the current approximate methods using sampled data. Further improvements and analysis of the centroid problem using root-finding and integral computation techniques are also possible.

The paper considers the analytical solution methods of the maximizing entropy or minimizing variance with fixed orness level problems and the maximizing orness with fixed entropy or variance value problems together. It proves that both of these two kinds of problems have common necessary conditions for their optimal solutions. The optimal solutions have the same forms and can be seen as the same OWA (ordered weighted averaging) weighting vectors from different points of view. The problems of minimizing orness problems with fixed entropy or variance constraints and their analytical solutions are proposed. Then these conclusions are extended to the corresponding RIM (regular increasing monotone) quantifier problems, which can be seen as the continuous case of OWA problems with free dimension. The analytical optimal solutions are obtained with variational methods.

The paper propose a general model that can combine the two kinds of OWA determination methods together: the optimization based methods and sample learning methods, that the relationship between these two kinds of problems is discussed. The analytical solution of this general model is proposed. Some properties of the problem are discussed. With this general model, we can observe the influences of the optimization criteria and the sample data on the OWA operator weight solution changes with the number of data examples.

The article proposes an extension of the BADD OWA operator—ANOWA (additive neat OWA) operator—and defines its orness measure. Some properties of the weighting function associated with orness level are analyzed. Then two special classes of ANOWA operator with maximum entropy and minimum variance are proposed, and the orness of the BADD OWA operator is discussed. For a given orness level, these ANOWA operators can be uniquely determined. Their aggregation values for any aggregation elements set always monotonically increase with their orness levels. Therefore they can be used as a parameterized aggregation method with orness as its control parameter and to represent the decision maker's preference. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 1045–1072, 2006.

In this paper, an orness measure to reflect the or-like degree of the quasi-arithmetic mean operator is proposed. With the generating function representation method, some properties of a quasi-arithmetic mean, associated with its orness measure, are analyzed. Then, the paper focuses on two kinds of parameterized quasi-arithmetic means with exponential function and power function generators respectively. Some properties of them are examined and compared. As the aggregation value for any set always monotonically increases with the orness level, the orness level can be used as a control parameter to represent the decision maker's preference. A generic method to get the quasi-arithmetic mean operator which makes the aggregation result consistent with its orness level is also examined. Finally, the properties of the quasi-arithmetic mean and that of the exponential and power function generators are extended to the weighted quasi-arithmetic mean by considering the importance of the aggregated elements

Based on the researches on ordered weighted average (OWA) operator, the weighted OWA operator (WOWA) and especially the quantifier guided aggregation method, with the generating function representation of regular increasing monotone (RIM) quantifier technique, we discuss the properties of WOWA operator with RIM quantifier in the respect of orness. With the continuous OWA and WOWA ideas recently proposed by Yager, an improvement on the continuous OWA and WOWA operator is proposed. The properties of WOWA are also extended from discrete to the continuous case. Based on these properties, two families of parameterized RIM quantifiers for WOWA operator are proposed, which have exponential generating function and piecewise linear generating function respectively. One interesting property of these two kinds of RIM quantifiers is that for any aggregated set (or variable) under any weighted (distribution) function, the aggregation values are always consistent with the orness (optimistic) levels, so they can be used to represent the decision maker's preference, and we can get the preference value of fuzzy sets or random variables with the orness level of RIM quantifier as their control parameter.

This article extends the quantifier-guided aggregation method to include probabilistic information. A general framework for the preference solution of decision making under an uncertainty problem is proposed, which can include decision making under ignorance and decision making under risk methods as special cases with some specific preference parameters. Almost all the properties, especially the monotonicity property, are kept in this general form. With the generating function representation of the Regular Increasing Monotone (RIM) quantifier, some properties of the RIM quantifier are discussed. A parameterized RIM quantifier to represent the valuation preference for probabilistic decision making is proposed. Then the risk attitude representation method is integrated in this quantifier-guided probabilistic decision making model to make it a general form of decision making under uncertainty. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 1253–1271, 2005.

By considering the decision maker's attitude of profit and risk, we propose an alternative selection method that can include the methods of decision making under ignorance and decision making under risk as special cases. An index to measure the decision maker's risk-averse degree is proposed. With a given optimistic level of profit and risk, the evaluation results of the alternatives can be obtained with a geometric ordered weighted average (OWA) operator and a basic defuzzification distribution (BADD) neat OWA operator. Some properties of these two kinds of OWA operator in the problem of decision making under uncertainty are also proposed. © 2004 Wiley Periodicals, Inc. Int J Int Syst 19: 1217–1238, 2004.