In the actual process of multi-attribute decision making (MADM), due to the complexity of objects and the inherent vagueness of human mind, the decision information is usually suitable to be expressed in natural language rather than a real number. However, natural languages always involve uncertainty and ambiguity, so it is difficult to avoid the loss of information in the process of decision making. The more the information loss, the less accurate results of decision are. In order to improve the accuracy of the decision making, it is necessary to correctly deal with linguistic decision information. And triangular norms, t-norms and s-norms and linguistic two-tuple are among the most effective ways to process linguistic information, and in this paper, based on Archimedean s-norm and linguistic two-tuple, some new operational laws of linguistic information are defined by using a continuous and strictly monotone increasing function and its inverse function. The prominent feature of such operations is that the operations are closed. Some main properties of these operations, such as commutativity, associativity and distribution law, are investigated. Moreover, considering the influence of expert weight on decision making, three new aggregation operators, including two-tuple linguistic extended Archimedean s-norm aggregation (TASTA) operator, two-tuple linguistic extended Archimedean s-norm weight averaging (TASTWA) operator and two-tuple linguistic vector extended Archimedean s-norm weight averaging (V-TASTWA) operator, are developed in this paper. All these aggregation operators use the consistency of group judgment to objectively adjust the expert weight and then effectively improve the accuracy of decision making. Later, a method for multi-attribute group decision making problems with two-tuple linguistic information is proposed based on TASTWA operator and V-TASTWA operator, and a numerical example is given to show its effectiveness and reasonability by comparison with other methods. The method not only overcomes the deficiency that the traditional operational laws of two-tuple linguistic information are not closed, but also makes full use of decision information to obtain the weight value and improves the accuracy and credibility of the results.