On the normalization of interval and fuzzy weights

The normalization of interval and fuzzy weights is often necessary in multiple criteria decision analysis (MCDA) under uncertainty, especially in analytic hierarchy process (AHP) with interval or fuzzy judgements. The existing normalization methods based on interval arithmetic and fuzzy arithmetic are found flawed and need to be revised. This paper presents the correct normalization methods for interval and fuzzy weights and offers relevant theorems in support of them. Numerical examples are examined to show the correctness of the proposed normalization methods and their differences from those existing normalization methods.