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With the changing business environment and the active participation of various stakeholders in the decision making process, it plays an increasingly important role to the weight of decision makers and the preference information given by decision makers. This paper presents a novel approach for group decision making under uncertainty with the involvement of the third-party evaluator in the decision making process. Recognizing the challenge in adequately determining the weight of decision makers in group decision making, the evidence theory is appropriately used with the involvement of the third-party evaluator. To effectively model the uncertainty and imprecision in the decision making process, fuzzy preference relations are used for better representing the subjective assessment of individual decision makers. To adequately determine the ranking of available alternatives, the logarithmic least square method is applied for appropriately aggregating the fuzzy preference relation of individual decision makers. A group consensus index is developed for facilitating consensus building in group decision making. This leads to better group decisions being made. A real-world application is presented that shows the proposed approach is effective in solving group decision making problems under uncertainty.

The challenge of complex multi-attribute large group decision-making (CMALGDM) is reflected from three perspectives: interrelated attributes, large group decision makers (DMs) and dynamic decision environments. However, there are few decision techniques that can address the three perspectives simultaneously. This paper proposes a random intuitionistic fuzzy factor analysis model, aiming to address the challenge of CMALGDM from the three perspectives. The proposed method effectively reduces the dimensionality of the original data and takes into account the underlying random environmental factors which may affect the performances of alternatives. The development of this method follows three steps. First, the random intuitionistic fuzzy variables are developed to deal with a hybrid uncertain situation where fuzziness and randomness co-exist. Second, a novel factor analysis model for random intuitionistic fuzzy variables is proposed. This model uses specific mappings or functions to define the way in which evaluations are affected by the dynamic environment vector through data learning or prior distributions. Third, multiple correlated attribute variables and DM variables are transformed into fewer independent factors by a two-step procedure using the proposed model. In addition, the objective classifications and weights for attributes and DMs are obtained from the results of orthogonal rotated factor loading. An illustrative case and detailed comparisons of decision results in different environmental conditions are demonstrated to test the feasibility and validity of the proposed method.

Bimatrix games are reconsidered under assumption that players are loss averse and payoffs are fuzzy, where reference points are given exogenously. The impact of loss aversion on bimatrix game is investigated by applying credibility theory. Three solution concepts of credibilistic loss aversion Nash equilibria (i.e., expected loss aversion Nash equilibrium, optimistic loss aversion Nash equilibrium and pessimistic loss aversion Nash equilibrium) and their existence theorems are proposed. The sufficient and necessary conditions are presented to find three equilibria. It is found that three credibilistic loss aversion Nash equilibria are equivalent if confidence level is equal to 0.5. Furthermore, an analysis on credibilistic loss aversion equilibria with respect to loss aversion is performed in a 2 × 2 bimatrix game with triangular fuzzy payoffs. It is found that with a higher probability a more loss-averse player receives a preferred payoff in the mixed strategy Nash equilibrium in some situations, but receives the second highest payoff in other situations. Finally, a case study is shown to illustrate the validity of the bimatrix game with triangular fuzzy payoffs developed in this paper.

The palm oil industry faces significant pressure because of the environmental problems from the plantation process and the palm oil milling process in their supply chain, which is due to the constraints of natural resources and the growing ecological awareness among customers. Reverse logistics (RL) has been considered as a systematic approach for the palm oil industry to improve the environmental impact. However, there are more barriers which cannot be detected in the RL. Therefore, this study proposes robust and flexible strategies for overcoming such barriers by focusing on identifying and ranking the solutions on RL adoption in Thailand’s palm oil industry. We use a fuzzy multi-criteria decision-making (MCDM) methods based on fuzzy analytical network process (ANP) and Vise Kriterijumska Optimizacija I Kompromisno Resenje (VIKOR) to prioritize related solutions and strategies. Firstly, fuzzy ANP is applied to identify the weighting of each barrier using pairwise comparison, while the final ranking of the solution on RL adoption is obtained through VIKOR. This model will help business analysis and supply chain managers formulate both short-term and long-term, flexible decision strategies for successfully managing and implementing RL adoption in the supply chain scenarios. In addition, sensitivity analysis is used to illustrate the method, and comparisons with other existing methods is also presented in this study.

As an extension of the classical averaging operators, Choquet integral has been shown a powerful tool for decision theory. In this paper, a method based on the generalized interval-valued intuitionistic fuzzy Choquet integrals w.r.t. the generalized interaction indices is proposed for multiattribute group decision making problems, where the importance of the elements is considered, and their interactions are reflected. Based on the given operational laws on interval-valued intuitionistic fuzzy sets, the interval-valued intuitionistic fuzzy Choquet integrals with respect to the generalized Shapley and Banzhaf indices are defined. Moreover, some of their properties are studied, such as idempotency, boundary, comonotonic linearity and μ–linearity. Furthermore, a decision procedure based on the proposed operators is developed for solving multi-attribute group decision making under interval-valued intuitionistic fuzzy environment. Finally, a numerical example is provided to illustrate the developed procedure.

Triangular fuzzy reciprocal preference relations (TFRPRs) are good choices to denote the decision makers' fuzzy preferences. Consistency analysis is a very important topic to preference relations that can avoid the illogical ranking order. However, all previous multiplicative consistency concepts for TFRPRs cannot well address this type of preference relations. To address this issue, this paper introduces a new multiplicative consistency concept for TFRPRs, which can be seen as a natural extension of Tanino's multiplicative consistency concept for reciprocal preference relations. Based on the new consistency concept, 0–1 mixed programming models for judging the multiplicative consistency of TFRPRs are constructed, and an approach for deriving multiplicative consistent TFRPRs is provided. Considering the incomplete case, programming models to determine missing values are constructed. Subsequently, a multiplicative consistency based method for decision making with TFRPRs is presented that can address the inconsistent and incomplete cases. Meanwhile, illustrative examples are provided to show the concrete application of the proposed procedure, and comparison analysis is also made.

Although many researchers have realized that the trade determination problem in one-shot multi-attribute exchanges is an important decision problem, few studies can be really applied to deal with this issue under a fuzzy environment. This paper proposes a multi-objective matching approach for one-shot multi-attribute exchanges with fuzzy information. First, we use fuzzy set theory to represent the traders’ orders with fuzzy information and then give a framework for the proposed multi-objective matching approach. Second, we present a novel calculation method of the matching degree based on the improved fuzzy information axiom. According to the matching degree, we build a fuzzy multi-objective optimization model (FMOM) for the trade determination problem, where two criteria, i.e., profits and satisfactions, are considered simultaneously. Afterward, in order to solve the FMOM, the fuzzy chance constrained programming technique based on credibility measures and the fuzzy goal programming method are introduced to convert it into a crisp single objective one. Finally, we conduct the numerical study and analysis on a practical example to illustrate the application and effectiveness of the proposed multi-objective matching approach.

We research a series of new aggregation operators based on Hamacher t-norm and t-conorm for hesitant fuzzy information.Some Hamacher operational rules of hesitant fuzzy sets are developed.A wide range of new hesitant fuzzy aggregation operators is proposed.We investigate their desirable properties and discuss the relationships between them.The proposed aggregation operators have more flexible for multicriteria decision making under hesitant fuzzy information environment. As a fuzzy set extension, the hesitant set is effectively used to model situations where it is allowable to determine several possible membership degrees of an element to a set due to the ambiguity between different values. We first introduce some new operational rules of hesitant fuzzy sets based on the Hamacher t-norm and t-conorm, in which a family of hesitant fuzzy Hamacher operators is proposed for aggregating hesitant fuzzy information. Some basic properties of these proposed operators are given, and the relationships between them are shown in detail. We further discuss the interrelations between the proposed aggregation operators and the existing hesitant fuzzy aggregation operators. Applying the proposed hesitant fuzzy operators, we develop a new technique for hesitant fuzzy multicriteria decision making problems. Finally, the effectiveness of the proposed technique is illustrated by mean of a practical example.

In this paper, another important solution for fuzzy games, called the Banzhaf function, is investigated. First, a sim- plified expression for the Banzhaf function for games with fuzzy coalition is proposed. The relationship between this simplified ex- pression and two kinds of particular Banzhaf functions are dis- cussed in detail. Furthermore, in the framework of games with fuzzy coalitions, we give a definition of the potential function of fuzzy games, and prove that the marginal values of the potential function are the components of the Banzhaf function. Further- more, a new axiomatic characterization of the Banzhaf function for games with fuzzy coalitions is provided.

In order to solve a problem of a discrete stochastic multiple-criteria decision making (MCDM) with aspiration levels, a method on the basis of prospect stochastic dominance is proposed in this paper. The psychological behavior of decision maker, for instance, judgmental distortion, reference dependence and loss aversion, are considered. Based on the concept of prospect theory, aspiration levels are initially took to be the reference points. Definition and related analysis of prospect stochastic dominance degree (PSDD) is given to describe the degree that one alternative dominates another when the prospect stochastic dominance relation for each pair of alternatives with respect to aspiration level is determined. On the basis of the PSDD matrix of alternative pairwise comparisons regarding the aspiration level of every criterion, an overall PSDD matrix is constructed using Choquet integral. Further, according to the concept of the PROMETHEE II, an outramking method is designed to collect the alternatives ranking result. Finally, the effectiveness and applicability of the method proposed are illustrated by two given numerical examples.

In this paper, by combining the Archimedean t-conorm and t-norm and Quasi-Choquet averaging operator, we develop an extended Atanassov's intuitionistic fuzzy Quasi-Choquet geometric operator to aggregate input arguments that are Atanassov's intuitionistic fuzzy values, where the inter-dependent or interactive characteristics among input arguments are taken into account. Some of the desirable properties and some special cases are investigated in detail. It is worth pointing out that most of the existing Atanassov's intuitionistic fuzzy geometric aggregation operators are special cases of this proposed aggregation operator. Furthermore, a decision procedure based on the proposed aggregation operator is developed for solving the multicriteria decision making problem in which all decision information is represented by Atanassov's intuitionistic fuzzy values. An illustrative example is given for demonstrating the applicability of the proposed decision procedure.

Fuzziness is inherent in decision data and decision making process. In this paper, interval-valued intuitionistic fuzzy set is used to capture fuzziness in multi-criteria decision making problems. The purpose of this paper is to develop a new method for solving multi-criteria decision making problem in interval-valued intuitionistic fuzzy environments. First, we introduce and discuss the concept of interval-valued intuitionistic fuzzy point operators. Using the interval-valued intuitionistic fuzzy point operators, we can reduce the degree of uncertainty of the elements in a universe corresponding to an interval-valued intuitionistic fuzzy set. Then, we define an evaluation function for the decision-making problem to measure the degrees to which alternatives satisfy and do not satisfy the decision-maker's requirement. Furthermore, a series of new score functions are defined for multi-criteria decision making problem based on the interval-valued intuitionistic fuzzy point operators and the evaluation function and their effectiveness and advantage are illustrated by examples.

In this study, we first define the uncertain linguistic Choquet averaging ULCA operator and the generalized Shapley uncertain linguistic Choquet averaging GS-ULCA operator. Especially, the GS-ULCA operator does not only globally consider the importance of the elements and their ordered positions, but also overall reflect the correlations among them and their ordered positions. Meantime, some desirable properties are studied. Furthermore, an approach to multi-attribute group decision making under uncertain linguistic environment is developed, where the interactions among experts and attributes are respectively considered. Moreover, if the information about the weights of experts and attributes is incompletely known, the models for the optimal fuzzy measures on expert set and attribute set are respectively established, which are based on gray relational analysis GRA method. Finally, a practical application of the developed approach to the problem of evaluating university faculty for tenure and promotion is provided.

The Shapley function is a very effective tool to measure the importance of elements, which can reflect the interactive characteristics among them. In this study we use the Shapley function to propose an induced generalized interval-valued intuitionistic fuzzy hybrid Shapley averaging (IG-IVIFHSA) operator. This operator does not only globally consider the importance of elements and their ordered positions, but also overall reflect the interaction among them and among their ordered positions. It is worth pointing out that most of the existing hybrid aggregation operators are special cases of our operator. Meantime, some important cases are considered, and some desirable properties are studied. Furthermore, the models for the optimal fuzzy measures on attribute set and ordered set are established, respectively. Moreover, an approach to multi-attribute decision making under interval-valued intuitionistic fuzzy environment is developed. Finally, two numerical examples are given to verify the developed method and demonstrate its practicality and feasibility.

Linguistic preference relation is a useful tool for expressing preferences of decision makers in group decision making according to linguistic scales. But in the real decision problems, there usually exist interactive phenomena among the preference of decision makers, which makes it difficult to aggregate preference information by conventional additive aggregation operators. Thus, to approximate the human subjective preference evaluation process, it would be more suitable to apply non-additive measures tool without assuming additivity and independence. In this paper, based on @l-fuzzy measure, we consider dependence among subjective preference of decision makers to develop some new linguistic aggregation operators such as linguistic ordered geometric averaging operator and extended linguistic Choquet integral operator to aggregate the multiplicative linguistic preference relations and additive linguistic preference relations, respectively. Further, the procedure and algorithm of group decision making based on these new linguistic aggregation operators and linguistic preference relations are given. Finally, a supplier selection example is provided to illustrate the developed approaches.

Yager (Fuzzy Sets, Syst 2003;137:59–69) extended the idea of order-induced aggregation to the Choquet aggregation and defined induced Choquet ordered averaging operator. In this paper, an induced intuitionistic fuzzy Choquet (IFC) integral operator is proposed for the multiple criteria decision making. Some of its properties are investigated. Furthermore, an induced generalized IFC integral operator is introduced. It is worth mentioning that most of the existing intuitionistic fuzzy aggregation operators are special cases of this induced aggregation operator. A decision procedure based on the proposed induced aggregation operator is developed for solving the multicriteria decision-making problem in which all the decision information is represented by intuitionistic fuzzy values. An illustrative example is given for demonstrating the applicability of the proposed decision procedure. © 2011 Wiley Periodicals, Inc. © 2011 Wiley Periodicals, Inc.

An extension of TOPSIS, a multi-criteria interval-valued intuitionistic fuzzy decision making technique, to a group decision environment is investigated, where inter-dependent or interactive characteristics among criteria and preference of decision makers are taken into account. To get a broad view of the techniques used, first, some operational laws on interval-valued intuitionistic fuzzy values are introduced. Based on these operational laws, a generalized interval-valued intuitionistic fuzzy geometric aggregation operator is proposed which is used to aggregate decision makers' opinions in group decision making process. In addition, some of its properties are discussed. Then Choquet integral-based Hamming distance between interval-valued intuitionistic fuzzy values is defined. Combining the interval-valued intuitionistic fuzzy geometric aggregation operator with Choquet integral-based Hamming distance, an extension of TOPSIS method is developed to deal with a multi-criteria interval-valued intuitionistic fuzzy group decision making problems. Finally, an illustrative example is used to illustrate the developed procedures.

For the real decision making problems, most criteria have inter-dependent or interactive characteristics so that it is not suitable for us to aggregate them by traditional aggregation operators based on additive measures. Thus, to approximate the human subjective decision making process, it would be more suitable to apply fuzzy measures, where it is not necessary to assume additivity and independence among decision making criteria. In this paper, an intuitionistic fuzzy Choquet integral is proposed for multiple criteria decision making, where interactions phenomena among the decision making criteria are considered. First, we introduced two operational laws on intuitionistic fuzzy values. Then, based on these operational laws, intuitionistic fuzzy Choquet integral operator is proposed. Moreover, some of its properties are investigated. It is shown that the intuitionistic fuzzy Choquet integral operator can be represented by some special t-norms and t-conorms, and it is also a generalization of the intuitionistic fuzzy OWA operator and intuitionistic fuzzy weighted averaging operator. Further, the procedure and algorithm of multi-criteria decision making based on intuitionistic fuzzy Choquet integral operator is given under uncertain environment. Finally, a practical example is provided to illustrate the developed approaches.

The inclusion variation and chain variation on T∞-tribes play an important role in T∞-measures and fuzzy games theory, where they were used for additive set functions on T∞-tribe. In this paper classical variation theory is extended to non-additive set functions on Ttribes, after defining the inclusion variation, disjoint and chain variation on T-tribes, we investigate some properties of these three variations in detail, such as, the (null-) null-additivity, exhaustivity, order continuity, continuity from the left and so on. An alternative proof of the Jordan decomposition theorem is given. Thus, some results pertaining to variations in classical measure theory are generalized. In some case, as for T∞-measures on T∞-tribe, its different formal variations can be extended to Ts-measures on Ts-tribes.

Prospect theory is a very effective method to express behavioral decision making under uncertainty. This paper attempts to develop a method to multi-attribute decision making with Atanassov's interval-valued intuitionistic fuzzy information using prospect theory. This method first transforms Atanassov's interval-valued intuitionistic fuzzy variables into the prospect values using the value function in prospect theory. Based on the aspiration levels, Atanassov's intuitionistic fuzzy prospect gain and loss matrices are obtained. Then, using the Atanassov's interval-valued intuitionistic hybrid weight averaging (IVIHWA) operator or the Atanassov's interval-valued intuitionistic hybrid Shapley weight averaging (IVIHSWA) operator, the comprehensive Atanassov's interval-valued intuitionistic fuzzy prospect value of each alternative is calculated. According to the comprehensive Atanassov's interval-valued intuitionistic fuzzy prospect values, a ranking method of alternatives is presented. Finally, two illustrative examples are selected to show the feasibility and availability of the proposed method.

Linguistic preference relation is a useful tool for expressing preferences of decision makers in group decision making according to linguistic scales. But in the real decision problems, there usually exist interactive phenomena among the preference of decision makers so that it is not suitable for us to aggregate preference information by conventional additive aggregation operators. Thus, to approximate the human subjective preference evaluation process, it would be more suitable to apply non-additive measures tool, where it is not necessary to assume additivity and independence among preference of decision makers. In this paper, based on the fuzzy measure, we developed a new linguistic ordered geometric averaging operator to aggregate the multiplicative linguistic preference relations, where interactions or dependence among subjective preference of decision makers are considered. Further, the procedure and algorithm of group decision making based on the new linguistic aggregation operators is given. Finally, a practical example is provided to illustrate the developed approaches.