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As a kind of uncertain theory, interval-valued numbers describe the overall values of ambiguous variables. And the conception of interval-valued number has been applied widely in fuzzy multiple attribute decision making because it is more suitable that some ambiguous attributes values are expressed in the form of interval-valued fuzzy numbers than in the form of certain numbers. Multiple attribute decision making methods are usually researched by interval-valued numbers which is one dimension. In this paper, new ranking decision making method is investigated by using the conception of multi-dimension interval-valued numbers. Interval-valued numbers and multiple attribute decision making methods are reviewed firstly. Then the conception of multiple-dimension interval-valued number is proposed and some properties of multiple-dimension interval-valued numbers are derived. Following new comparing criteria of different interval-valued numbers is discussed. And new order relationship of the variables in multiple attribute decision making is given by the ranking comparing criteria. Finally, an example is performed to illustrate the proposed method.