In three-way group decision-making based on the minimum risk Bayesian decision theory, the consensus of basic loss function evaluation becomes its main core issue. However, if we only consider evaluation information consensus, it does not ensure the classification quality of three-way decisions. Thus, to balance the consensus and decision quality, we design a three-way group decision-making joint learning process via constructing a two-stage group consensus method. Inspired by supervised learning, Stage 1 establishes the minimum decision error optimization model (MDEOM) to learn the optimal parameters of three-way decisions and calculate decision loss reference values. Then, we design an algorithm to solve MDEOM based on particle swarm optimization (PSO) algorithm. In Stage 2, we calculate adjusted decision losses with the minimum decision loss difference consensus model (MDLDCM), which can guide the consensus adjustment of loss functions and improve the decision quality of three-way group decision making. Finally, some implications of three-way group decision making with the two-stage group consensus method are discussed by a series of experiments which prove the improvement effectiveness of our proposed method.

The purpose of this article is to present a novel biobjective biform game framework for optimizing strategy selection of nodes (agents) in bilateral link network formation under dual objectives of profit and link security. First, in the first part of the noncooperative game, we construct various possible strategy combinations (situations) of all players (nodes) in the network, which define the competitive environment in the second part of the cooperative game. Second, in the second part, we derive coalitions’ payoffs under the two objectives of profit and link security and propose the biobjective Banzhaf values to allocate the corresponding cooperative games in all strategy situations. Then, a biobjective noncooperative game is formed on the basis of the obtained biobjective Banzhaf values. Third, we redefine Pareto–Nash equilibriums (in the sense of pure strategies) and obtain the Pareto–Nash equilibriums of the biobjective noncooperative game in the first part. Subsequently, the choice of the obtained Pareto–Nash equilibriums is regarded as a biobjective group decision-making problem, which is solved by employing the redefined Borda choice function method with objective weights. Hence, we can obtain the optimal strategy choices of all the nodes (players) for the entire bilateral link network formation. Finally, numerical examples and comparative analysis are given to verify the validity and applicability of the proposed models and methods.

Rough interval (RI) is an appropriate generalization of the crisp interval and is very much useful to express uncertain parameters or partially unknown variables when they contain dual-layer information. In real-life problems, there are many situations where players of a matrix game cannot assess their pay-offs by using fuzzy sets/intuitionistic fuzzy sets or ordinary intervals. RIs are used as an excellent tool to handle such situations. This paper explores matrix games with RI pay-offs and investigates two different solution methodologies to solve such a game. In the first approach, a pair of auxiliary linear programming problems with RI coefficients for the players has been constructed. Then each of the two auxiliary programming problems is converted into two linear programming problems with interval coefficients (LPPICs) using lower and upper approximation intervals of the RI. Finally from each LPPIC, two classical linear programming problems (LPPs) are constructed. In the second approach, the expected value technique for RI is used for transforming auxiliary mathematical programming models under the RI environment to crisp LPP. A case study on the telecom market share problem is considered to show the applicability of the proposed approaches and results are compared and analyzed with an existing method.

The primary goal of this article is to develop a direct and simplified approach for computing triangular fuzzy Banzhaf values for a special class of cooperative games with coalitions' values represented by triangular fuzzy numbers, which are called triangular fuzzy cooperative games for short. Combining with the β-cut sets of coalitions' triangular fuzzy values, in this article, we can obtain a series of associated interval-valued cooperative games with the triangular fuzzy cooperative games. Hereby, we construct associated cooperative games with the interval-valued cooperative games. It is proven that the Banzhaf values of the associated cooperative games with interval-valued cooperative games are monotonic and nondecreasing functions under some size monotonicity-like conditions. Hence, the mean and the lower and upper limits of the triangular fuzzy Banzhaf values can be directly and explicitly attained through using only the mean, and the lower and upper limits of the coalitions' triangular fuzzy values, respectively. The developed approach does not use the subtraction of triangular fuzzy numbers and, hereby, can effectively avoid the irrational issues resulting from it. Furthermore, some important properties of the triangular fuzzy Banzhaf values are proven and the applicability and validity of the developed approach are illustrated with a numerical example.

The purpose of this paper is to develop a compromise-typed variable weight decision method for solving hybrid multiattribute decision making problems with multiple types of attribute values. The compromise-typed variable weight functions are defined and constructed by utility functions. Moreover, the variable weight synthesis and the orness measures based on the coefficients of absolute risk aversion are analyzed in variable weight decision making. The comprehensive values of alternatives based on the compromise-typed variable weight decision method are calculated. The decision-making results are determined according to the comprehensive values. Finally, an example and a detailed comparison analysis are presented to show the applicability and validity of the proposed method.

There are some mistakes in the computation results of the real example in the article by Li, “TOPSIS-based nonlinear-programming methodology for multi-attribute decision making with interval-valued intuitionistic fuzzy sets” [IEEE Trans. Fuzzy Syst., vol. 18, no. 2, pp. 299–311, 2010], and this article provides corrections to that paper.

The aim of this paper is to develop a new methodology for solving matrix games with payoffs of Atanassov's intuitionistic fuzzy (I-fuzzy) numbers. In this methodology, we define the concepts of I-fuzzy numbers and the value-index and ambiguity-index and develop a difference-index-based ranking method, which is proven to be a total order. By doing this, the parameterized nonlinear programming models are derived from a pair of auxiliary I-fuzzy mathematical programming models, which are used to determine solutions of matrix games with payoffs of I-fuzzy numbers. The validity and applicability of the models and method proposed in this paper are illustrated with a practical example.

The aim of this paper is to develop a new Atanassov's intuitionistic fuzzy (A-IF) programming method to solve heterogeneous multiattribute group decision-making problems with A-IF truth degrees in which there are several types of attribute values such as A-IF sets (A-IFSs), trapezoidal fuzzy numbers, intervals, and real numbers. In this method, preference relations in comparisons of alternatives with hesitancy degrees are expressed by A-IFSs. Hereby, A-IF group consistency and inconsistency indices are defined on the basis of preference relations between alternatives. To estimate the fuzzy ideal solution (IS) and weights, a new A-IF programming model is constructed on the concept that the A-IF group inconsistency index should be minimized and must be not larger than the A-IF group consistency index by some fixed A-IFS. An effective method is developed to solve the new derived model. The distances of the alternatives to the fuzzy IS are calculated to determine their ranking order. Moreover, some generalizations or specializations of the derived model are discussed. Applicability of the proposed methodology is illustrated with a real supplier selection example.

Of the different types of games, the matrix games with fuzzy payoffs have been extensively discussed. Two major kinds of solution methods have been devised. One is the defuzzification approach based on ranking functions. Another is the two-level linear programming method which can obtain membership functions of players' fuzzy values (or gain floor and loss ceiling). These methods cannot always ensure that players' fuzzy/defuzzified values have a common value. The aim of this paper is to develop an effective methodology for solving matrix games with payoffs expressed by trapezoidal fuzzy numbers (TrFNs). In this methodology, we introduce the concept of Alpha-matrix games and prove that players' fuzzy values are always identical, and hereby, any matrix game with payoffs expressed by TrFNs has a fuzzy value, which is also a TrFN. The upper and lower bounds of any Alpha-cut of the fuzzy value and the players' optimal strategies are easily obtained through solving the derived four linear programming problems with the upper and lower bounds of Alpha-cuts of the fuzzy payoffs. In particular, the fuzzy value can be explicitly estimated through solving the auxiliary linear programming with data taken from the 1-cut and 0-cut of the fuzzy payoffs. The proposed method in this paper is illustrated with a real example and compared with other methods to show validity and applicability.

The aim of this paper is to develop a new methodology for solving group decision making problems in which preference comparisons between alternatives are expressed with Atanassov's intuitionistic fuzzy (IF) preference relations. In this methodology, the generalized ordered weighted averaging (GOWA) operator is extended to develop the Atanassov's IF set (IFS) generalized ordered weighted averaging (IFSGOWA) operator, which can aggregate vague or imprecise information expressed with the Atanassov's IFSs. The Atanassov's IFSGOWA operator based methodology is further developed to solve group decision making problems with Atanassov's IF preference relations. A real example of the agroecological region assessment problem is used to illustrate the implementation and applicability of the proposed methodology. It is also shown that the obtained decision results could be affected by the choice of the parameter lambda and the nature of the weights.

Interval-valued intuitionistic fuzzy (IVIF) sets are useful to deal with fuzziness inherent in decision data and decision-making processes. The aim of this paper is to develop a nonlinear-programming methodology that is based on the technique for order preference by similarity to ideal solution to solve multiattribute decision-making (MADM) problems with both ratings of alternatives on attributes and weights of attributes expressed with IVIF sets. In this methodology, nonlinear-programming models are constructed on the basis of the concepts of the relative-closeness coefficient and the weighted-Euclidean distance. Simpler auxiliary nonlinear-programming models are further deduced to calculate relative-closeness of IF sets of alternatives to the IVIF-positive ideal solution, which can be used to generate the ranking order of alternatives. The proposed methodology is validated and compared with other similar methods. A real example is examined to demonstrate the applicability and validity of the methodology proposed in this paper.

The purpose of this paper is to develop the concept and mathematical-programming methodology of matrix games with payoffs represented by Atanassov's interval-valued intuitionistic fuzzy (IVIF) sets. In this methodology, the concept of solutions of matrix games with payoffs represented by Atanassov's IVIF sets is defined, and some important properties are studied using multiobjective-programming and duality-programming theory. It is proven that each matrix game with payoffs represented by Atanassov's IVIF sets has a solution, which can be obtained through solving a pair of auxiliary linear/nonlinear-programming models derived from a pair of nonlinear biobjective interval-programming models. Validity and applicability of the proposed methodology are illustrated with a numerical example.

The aim of this paper is to use the concept of extended linguistic variables to develop a new methodology for solving multiattribute group decision making problems under linguistic assessments. In this methodology, using the generalized induced ordered weighted averaging operator and extended linguistic variables, linguistic ratings of each alternative on all attributes for each decision maker are aggregated into a comprehensive assessment, which is not always a linguistic term in the linguistic term set designated a priori. Comprehensive assessments of each alternative on all decision makers are similarly aggregated into a comprehensive assessment of the alternative for the group. An optimal linguistic term in the linguistic term set which approaches the comprehensive assessment of each alternative for the group as close as possible is computed and an extended 2-tuple fuzzy linguistic representation is obtained for the alternative. The ranking order of alternatives for the group is generated according to the extended 2-tuple fuzzy linguistic representations. A numerical example is examined to demonstrate applicability and implementation process of the methodology proposed in this paper.

The intuitionistic fuzzy set (IFS) introduced by Atanassov in 1986 is characterized by two functions and has rarely been applied to the game theory yet. The aim of this paper is to develop the concept and methodology of matrix games with payoffs of Atanassov's IFSs. In this methodology, the concepts of the solutions for matrix games with payoffs of Atanassov's IFSs are defined based on the degree of membership and the degree of non-membership. It is proven that matrix games with payoffs of Atanassov's IFSs have solutions, which can be obtained through solving a pair of nonlinear programming models derived from two auxiliary nonlinear bi-objective programming models. The proposed method is illustrated with a numerical example.

The purpose of the paper is to study how to solve a type of matrix games with payoffs of triangular fuzzy numbers. In this paper, the value of a matrix game with payoffs of triangular fuzzy numbers has been considered as a variable of the triangular fuzzy number. First, based on two auxiliary linear programming models of a classical matrix game and the operations of triangular fuzzy numbers, fuzzy optimization problems are established for two players. Then, based on the order relation of triangular fuzzy numbers the fuzzy optimization problems for players are decomposed into three-objective linear programming models. Finally, using the lexicographic method maximin and minimax strategies for players and the fuzzy value of the matrix game with payoffs of triangular fuzzy numbers can be obtained through solving two corresponding auxiliary linear programming problems, which are easily computed using the existing Simplex method for the linear programming problem. It has been shown that the models proposed in this paper extend the classical matrix game models. A numerical example is provided to illustrate the methodology.

There exists little investigation on multiattribute decision making under intuitionistic fuzzy environments although both crisp and fuzzy multiattribute decision making have achieved a great progress. In this paper, multiattribute decision making problems using intuitionistic fuzzy sets are investigated and the TOPSIS is further extended to develop one new methodology for solving such problems. In this methodology, an interval fractional programming model is constructed on the basis of the relative closeness coefficient using the TOPSIS. Comprehensive evaluation of each alternative, which may be described as an intuitionistic fuzzy set or interval number, is calculated using two auxiliary mathematical programming problems derived from the interval fractional programming model proposed in this paper. Optimal degrees of membership for alternatives are calculated to determine their ranking order using the concept of likelihood based on the ranking method of interval numbers. Implementation process of the method proposed in this paper is illustrated with a numerical example.

The aim of this paper is to develop a fuzzy linear programming technique for multidimensional analysis of preference (FLINMAP) in multiattribute group decision making problems with linguistic variables and incomplete preference information. In this paper, linguistic variables are used to assess an alternative on qualitative attributes using fuzzy ratings corresponding to some triangular fuzzy numbers. Each alternative is assessed on the basis of its distance to a fuzzy positive ideal solution (FPIS) which is unknown a priori. The FPIS and the weights of attributes are calculated by constructing a new linear programming model based on the group consistency and inconsistency indices defined on the basis of preferences between alternatives given by the decision makers. The distance of each alternative to the FPIS can be calculated to determine the ranking order of all alternatives. The implementation process of this methodology is demonstrated with an example.

The purpose of the paper is to introduce a new type of fuzzy matrix games: fuzzy constrained matrix games. A computational method for its solution based on establishment of the auxiliary fuzzy linear programming for each player is proposed. The approach based on the multiobjective programming is establisched to solve these fuzzy linear programming. Effectiveness is illustrated with a numerical example.