Deng-Feng Li

Corrections to “TOPSIS-Based Nonlinear-Programming Methodology for Multi-attribute Decision Making With Interval-Valued Intuitionistic Fuzzy Sets”

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.

Mathematical-Programming Approach to Matrix Games With Payoffs Represented by Atanassov's Interval-Valued Intuitionistic Fuzzy Sets

The purpose of this paper is to develop the concept and mathematical-programming methodology of matrix games with payoffs represented by Atanassovs interval-valued intuitionistic fuzzy (IVIF) sets. In this methodology, the concept of solutions of matrix games with payoffs represented by Atanassovs 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 Atanassovs 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.

Multiattribute decision making method based on generalized OWA operators with intuitionistic fuzzy sets

The intuitionistic fuzzy (IF) set characterized by two functions was a generalization of the fuzzy set. In this paper, we investigate multiattribute decision making (MADM) problems with ratings of alternatives being expressed using IF sets and attribute weights given as real numbers. Firstly, the generalized ordered weighted averaging (GOWA) operators introduced by Yager [Yager, R. (2004). Generalized OWA aggregation operators. Fuzzy Optimization and Decision Making, 3, 93-107] are extended to aggregate IF sets. Secondly, MADM problems with IF sets are formulated through transforming the ratings of alternatives on both qualitative and quantitative attributes into IF sets in a unified way. The method and procedure based on the extended GOWA operators are developed to solve the MADM problems with IF sets. Finally, the effectiveness and practicability of the proposed method are illustrated with a numerical example.

Atanassov's Intuitionistic Fuzzy Programming Method for Heterogeneous Multiattribute Group Decision Making With Atanassov's Intuitionistic Fuzzy Truth Degrees

The aim of this paper is to develop a new Atanassovs 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.

A systematic approach to heterogeneous multiattribute group decision making

Heterogeneous multiattribute group decision making (MAGDM) problems which involve multi-granularity linguistic labels, fuzzy numbers, interval numbers and real numbers are very complex and important in practical applications of decision making theory. Hitherto, there exists no general theoretical inducement for solving such problems. The purpose of this paper is to develop a systematic methodology for solving the heterogeneous MAGDM problems by introducing the multiattribute ranking index based on the particular measure of closeness to the positive ideal solution (PIS) and using the weighted Minkowski distance to measure differences between each alternative and the PIS as well as the negative ideal solution (NIS). The proposed methodology is shown to have some advantages over the fuzzy TOPSIS. Validity and applicability of the methodology proposed in this paper is illustrated with a real example of the missile weapon system selection problem.

Interval programming models for matrix games with interval payoffs

The aim of this paper is to study how to solve a type of matrix game with interval payoffs. In this paper, the interval inequality relations and the concept of solutions of the matrix game with interval payoffs are defined. Based on the fuzzy ranking index defined, the solution of the matrix game with interval payoffs can be obtained through solving a pair of bi-objective linear programming models derived from the constructed auxiliary interval programming models. It is shown that the models proposed in this paper extend those of the classical matrix game. The validity and applicability of the proposed methodology are illustrated with a numerical example.

A NONLINEAR PROGRAMMING APPROACH TO MATRIX GAMES WITH PAYOFFS OF ATANASSOV'S INTUITIONISTIC FUZZY SETS

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 Atanassovs IFSs. In this methodology, the concepts of the solutions for matrix games with payoffs of Atanassovs IFSs are defined based on the degree of membership and the degree of non-membership. It is proven that matrix games with payoffs of Atanassovs 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.

A Parameterized Nonlinear Programming Approach to Solve Matrix Games With Payoffs of I-Fuzzy Numbers

The aim of this paper is to develop a new methodology for solving matrix games with payoffs of Atanassovs 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.

Multi-attribute decision making method considering the amount and reliability of intuitionistic fuzzy information

Imprecision and vagueness often occur in practical multi-attribute decision making MADM problems. Intuitionistic fuzzy IF sets are flexible to simulate these situations. The aim of this paper is to develop an effective method for solving MADM problems in which the attribute values are expressed with IF sets. Inspired by TOPSIS, we propose a new ranking function of IF sets, which takes into the amount and the reliability of an IF set. Hereby we develop a new MADM method. An example of the investment selection problem is examined to demonstrate applicability and feasibility of the proposed method. It is shown that the proposed method has some advantages over other methods.

Possibility mean and variance based method for multi-attribute decision making with triangular intuitionistic fuzzy numbers

Triangular intuitionistic fuzzy numbers TIFNs are useful to deal with ill-known quantities in decision making problems. The focus of this paper is on multi-attribute decision making MADM problems in which the attribute values are expressed with TIFNs and the information on attribute weights is incomplete, which are solved by developing a new decision method based on possibility mean and variance of TIFNs. The notions of possibility mean and variance for TIFNs are introduced as well as the possibility standard deviation. A new ranking approach for TIFNs is developed according to the ratio of the possibility mean to the possibility standard deviation. Hereby we construct a bi-objective programming model, which maximizes the ratios of the possibility mean to the possibility standard deviation for membership and non-membership functions on alternatives overall attribute values. Using the lexicographic approach, the bi-objective programming model is transformed into two non-linear programming models, which are further transformed into the linear programming models by using the variable transformation. Thus, we can obtain the maximum ratios of the possibility mean to the possibility standard deviation, s are used to rank the alternatives. A numerical example is examined to demonstrate applicability and implementation process of the proposed method.