The traditional approach for the problems of sorting permutations by rearrangements is to consider that all operations have the same unitary cost. In this case, the goal is to find the minimum number of allowed rearrangements that are needed to sort a given permutation, and numerous efforts have been made over the past years regarding these problems. On the other hand, a long rearrangement (which is in fact a mutation) is more likely to disturb the organism. Therefore, weights based on the length of the segment involved may have an important role in the evolutionary process. In this paper we present the first results regarding problems of sorting permutations by length-weighted operations that consider rearrangement models with prefix and suffix variations of reversals and transpositions, which are the two most common types of genome rearrangements. Our main results are O(lg2⁡n)O(lg2⁡n)-approximation algorithms for 10 such problems.