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The significant characteristic of the TODIM (an acronym in Portuguese of Interactive and Multiple Attribute Decision Making) method is that it can consider the bounded rationality of the decision makers. However, in the classical TODIM method, the rating of the attributes only can be used in the form of crisp numbers. Because 2-dimension uncertain linguistic variables can easily express the fuzzy information, in this article, we extend the TODIM method to 2-dimension uncertain linguistic information. First of all, the definition, characteristics, expectation, comparative method and distance of 2-dimension uncertain linguistic information are introduced, and the steps of the classical TODIM method for Multiple attribute decision making (MADM) problems are presented. Second, on the basis of the classical TODIM method, the extended TODIM method is proposed to deal with MADM problems in which the attribute values are in the form of 2-dimension uncertain linguistic variables, and detailed decision steps are given. Its significant characteristic is that it can fully consider the bounded rationality of the decision makers, which is a real action in real decision making. Finally, a numerical example is provided to verify the developed approach and its practicality and effectiveness.

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The power average (PA) operator and Maclaurin symmetric mean (MSM) operator are two important tools to handle the multiple attribute group decision‐making (MAGDM) problems, and the combination of two operators can eliminate the influence of unreasonable information from biased decision makers (DMs) and can capture the interrelationship among any number of arguments. The Pythagorean fuzzy linguistic set (PFLS) is parallel to the intuitionistic linguistic set (ILS), which is more powerful to convey the uncertainty and ambiguity of the DMs than ILS. In this paper, we propose some power MSM aggregation operators for Pythagorean fuzzy linguistic information, such as Pythagorean fuzzy linguistic power MSM operator and Pythagorean fuzzy linguistic power weighted MSM (PFLPWMSM) operator. At the same time, we further discuss the properties and special cases of these operators. Then, we propose a new method to solve the MAGDM problems with Pythagorean fuzzy linguistic information based on the PFLPWMSM operator. Finally, some illustrative examples are utilized to show the effectiveness of the proposed method.

On the basis of the normal intuitionistic fuzzy numbers (NIFNs), we proposed the normal interval-valued intuitionistic fuzzy numbers (NIVIFNs) in which the values of the membership and nonmembership were extended to interval numbers. First, the definition, the properties, the score function and accuracy function of the NIVIFNs are briefly introduced, and the operational laws are defined. Second, some aggregation operators based on the NIVIFNs are proposed, such as normal interval-valued intuitionistic fuzzy weighted arithmetic averaging operator, normal interval-valued intuitionistic fuzzy ordered weighted arithmetic averaging operator, normal interval-valued intuitionistic fuzzy hybrid weighted arithmetic averaging operator, normal interval-valued intuitionistic fuzzy weighted geometric averaging operator, normal interval-valued intuitionistic fuzzy ordered weighted geometric averaging operator, normal interval-valued intuitionistic fuzzy hybrid weighted geometric averaging operator, and normal interval-valued intuitionistic fuzzy generalized weighted averaging operator, normal interval-valued intuitionistic fuzzy generalized ordered weighted averaging operator, normal interval-valued intuitionistic fuzzy generalized hybrid weighted averaging operator, and some properties of these operators, such as idempotency, monotonicity, boundedness, commutativity, are studied. Further, an approach to the decision making problems with the NIVIFNs is established. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness. V C 2015 Wiley Periodicals, Inc. Complexity 21: 277-290, 2016 Key Words: multicriteria decision making; normal interval-valued intuitionistic fuzzy numbers; normal interval-valued intuitionistic fuzzy aggregation operator C O M P L E X I T Y 277 Q

Normal neutrosophic numbers (NNNs) are an important tool to describe the decision making problems, and they are more appropriate to express the incompleteness, indeterminacy and inconsistency of the evaluation information. In this paper, we firstly introduce the definition, the properties, the score function, the accuracy function, and the operational laws of the NNNs. Then, some operators are proposed, such as the normal neutrosophic power averaging operator, the normal neutrosophic weighted power averaging operator, the normal neutrosophic power geometric operator, the normal neutrosophic weighted power geometric operator, the normal neutrosophic generalized power averaging operator, the normal neutrosophic generalized weighted power averaging (NNGWPA) operator. Furthermore, some properties of them are discussed. Thirdly, we propose a multiple attribute decision making method based on the NNGWPA operator. Finally, we use an illustrative example to demonstrate the practicality and effectiveness of the proposed method.

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