In the current study a novel TODIM (an acronym in Portuguese of Interactive and Multi-criteria Decision Making) method has been proposed to solve multiple criteria decision making problems in which the decision matrix is characterised by general grey numbers. Firstly, a new axiom definition of distance measure is defined and two distance formulae are proposed. Later, a new standardized method for general grey numbers is put forward. Based on the distance formulae, a grey incidence method considering position weights using the idea of OWA operator is presented. Furthermore, a new approach to obtain the criteria weights using the grey incidence method is proposed. Moreover, a new TODIM method with general grey numbers, considering the bounded rationality of decision-maker is presented. Finally, a realistic case illustrates the effectiveness of the proposed methodology.