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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. [ABSTRACT FROM AUTHOR]

In many real games, two players' payoffs are not exactly opposite and players often have some constraints or preference on their strategies. Such kinds of games are called constrained bi-matrix games (CBGs) for short. Based on dual programming theory, two linear programming models are developed for solving any CBG. Then, a classic example of bi-matrix games called the Rock-scissors-cloth game with considering players' preference on strategies is used to show the validity of the proposed models and method. Furthermore, we investigate on the CBGs with payoffs represented by intuitionistic fuzzy numbers, which are simply called intuitionistic fuzzy CBGs in which both the ambiguity of the payoffs and the constraints of the strategies are taken into account. At last, the effectiveness of the proposed models and method is demonstrated with a numerical example of the company development strategy choice problem. [ABSTRACT FROM AUTHOR]

Owing to more vague concepts frequently represented in decision data, intuitionistic fuzzy sets (IFSs) are more flexibly used to model real-life decision situations. At the same time, with ever increasing complexity in many decision situations in reality, there are often some challenges for a decision maker to provide complete attribute preference information, i.e., the weights may be completely unknown or partially known. The aim of this paper is to develop an effective method for solving intuitionistic fuzzy multi-attribute decision making (MADM) problems with incomplete weight information. In this method, ratings of alternatives on attributes are expressed with IFSs. The multi-objective programming models are established to calculate unknown weights by using weight information partially known a priori. The derived minimum weighted Minkowski distance power models are used to determine the unknown weights and to generate the ranking order of the alternatives simultaneously. The proposed models are easily extended to intuitionistic fuzzy MADM problems with different weight information structures. An example of the supplier selection problem is examined to demonstrate applicability and flexibility of the proposed models and method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to develop a new ranking methodology for solving matrix games with intuitionistic fuzzy payoffs in terms of a value and ambiguity of triangular intuitionistic fuzzy numbers (TIFNs). In order to obtain an equilibrium solution of this matrix game, we construct two auxiliary TIFN programming models that are transformed into bi-objective linear programming models which are solved by a lexicographic methodology. The validity and applicability of the proposed methodology in this paper are illustrated with a numerical example. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

Highlights: [•] New fuzzy mathematical programming model is derived for complex MADM. [•] A possibility method is developed to solve the new fuzzy programming model. [•] A fuzzy LINMAP is developed to solve MADM with multiple types of attribute values. [ABSTRACT FROM AUTHOR]

Interval-valued intuitionistic fuzzy (IVIF) sets are a useful tool to deal with fuzziness inherent in decision data and decision making process. The aim of this paper is to develop a methodology for solving multiattribute decision making (MADM) with both ratings of alternatives on attributes and weights being expressed with IVIF sets. In this methodology, a weighted Euclidean distance between IF sets is defined using weights of IF sets. A pair of nonlinear programming models is constructed based on the concept of the relative closeness coefficients and the distance defined. Two simpler auxiliary nonlinear programming models are further derived to calculate the relative closeness coefficient intervals of alternatives to the IVIF positive ideal solution, which can be used to generate ranking order of alternatives based on the concept of likelihood of interval numbers. The method proposed in this paper is illustrated with a real example. [ABSTRACT FROM AUTHOR]

The aim of this note is to point out and correct some vital mistakes in the paper by P K Nayak and M Pal, "Linear programming technique to solve two person matrix (games with interval pay-offs). Asia-Pacific Journal of Operational Research, 26(2), 285-305". Lots of serious mistakes on the definitions, conclusions, models, methods, proofs and computing results have been corrected and modified in this note. We also indicate inappropriate formulations regarding their proposed linear programming models for solving generic matrix games with interval pay-offs and suggest a pair of linear programming models with any minimal acceptance degree of the interval inequality constraints which may be allowed to violate. The lexicographic method is suggested so that a rational and credible solution of the generic matrix game with interval pay-offs can be achieved. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to develop a closeness coefficient based nonlinear programming method for solving multiattribute decision making problems in which ratings of alternatives on attributes are expressed using interval-valued intuitionistic fuzzy (IVIF) sets and preference information on attributes is incomplete. In this methodology, nonlinear programming models are constructed on the concept of the closeness coefficient, which is defined as a ratio of the square of the weighted Euclidean distance between an alternative and the IVIF negative ideal solution (IVIFNIS) to the sum of the squares of the weighted Euclidean distances between the alternative and the IVIF positive ideal solution (IVIFPIS) as well as the IVIFNIS. Simpler nonlinear programming models are deduced to calculate closeness intuitionistic fuzzy sets of alternatives to the IVIFPIS, which are used to estimate the optimal degrees of membership and hereby generate ranking order of the alternatives. The derived auxiliary nonlinear programming models are shown to be flexible with different information structures and decision environments. The proposed method is validated and compared with other methods. A real example is examined to demonstrate applicability of the proposed method in this paper. [Copyright &y& Elsevier]

The purpose of this paper is to develop a nonlinear programming model and method for solving fuzzy multi-attribute group decision making (FMAGDM) problems with linguistic variables. In this methodology, alternatives are assessed on qualitative attributes using linguistic terms from the designated linguistic term set, which can be expressed with the triangular fuzzy number set designated a priori. Alternatives for decision makers are assessed on the basis of the distances between the ordered weighted averaging (OWA) comprehensive value and the group best alternative which is unknown a priori. The group best alternative is then estimated through solving an auxiliary nonlinear programming model based on the concept that the sum of the distances of alternatives from the group best alternative for the decision makers should be as small as possible. The obtained group best alternative can be expressed using the designated triangular fuzzy numbers, which are corresponding to the designated linguistic terms. A numerical example is examined to demonstrate the difference between the proposed method and the generalized induced OWA operator method. [ABSTRACT FROM AUTHOR]

Abstract: Fuzziness is inherent in decision data and decision making process. In this paper, interval-valued intuitionistic fuzzy (IVIF) sets are used to capture fuzziness in multiattribute decision making (MADM) problems. The purpose of this paper is to develop a methodology for solving MADM problems with both ratings of alternatives on attributes and weights being expressed with IVIF sets. In this methodology, a weighted absolute distance between IF sets is defined using weights of IF sets. Based on the concept of the relative closeness coefficients, we construct a pair of nonlinear fractional programming models which can be transformed into two simpler auxiliary linear programming models being used to calculate the relative closeness coefficient intervals of alternatives to the IVIF positive ideal solution, which can be employed to generate ranking order of alternatives based on the concept of likelihood of interval numbers. The proposed method is illustrated with a real example. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

Abstract: The Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) developed by Srinivasan and Shocker [V. Srinivasan, A.D. Shocker, Linear programming techniques for multidimensional analysis of preference, Psychometrika 38 (1973) 337–342] is one of the existing well-known methods for multiattribute decision making (MADM) problems. However, the LINMAP only can deal with MADM problems in crisp environments. Fuzziness is inherent in decision data and decision making processes, and linguistic variables are well suited to assessing an alternative on qualitative attributes using fuzzy ratings. The aim of this paper is further extending the LINMAP method to develop a new methodology for solving MADM problems under fuzzy environments. In this methodology, linguistic variables are used to capture fuzziness in decision information and decision making processes by means of a fuzzy decision matrix. A new vertex method is proposed to calculate the distance between trapezium fuzzy number scores. Consistency and inconsistency indices are defined on the basis of preferences between alternatives given by the decision maker. Each alternative is assessed on the basis of its distance to a fuzzy positive ideal solution (FPIS) which is unknown. The FPIS and the weights of attributes are then estimated using a new linear programming model based upon the consistency and inconsistency indices defined. Finally, the distance of each alternative to the FPIS can be calculated to determine the ranking order of all alternatives. A numerical example is examined to demonstrate the implementation process of this methodology. Also it has been proved that the methodology proposed in this paper can deal with MADM problems under not only fuzzy environments but also crisp environments. [Copyright &y& Elsevier]

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. [ABSTRACT FROM PUBLISHER]