Normalization approaches for multiplicative interval weights

The problem of interval weights normalization is of perennial interest, as it is related to the problems of Multiple Criteria Decision Making (MCDM). Existing normalization approaches are just taken into account one conventional situation that the product of some weight values is equal to 1, while ignoring the other situation, in which weights variables required to be multiplied to 1, i.e. multiplicatively normalization. This paper investigates normalization approaches for multiplicative interval weights. A pair of auxiliary models and two equivalent pairs of normalization approaches for multiplicative interval weights is developed respectively; relevant theorems are given to support them. Finally, a numerical example is furnished to illustrate the proposed approaches and demonstrate their effectiveness and practicability.