Before revision:  Here we achieve an improved approximation ratio of (3-\frac{3r{b(\pi)+r}), where r is the }  number of prefix reversals that must happen in the sorting, when possible, and b(\pi)\geq 2r is the number of breakpoints in the given permutation. Again, in this variation our algorithm performs close to 1.6 in practice .
After revision:  Here we achieve an improved approximation ratio of (3-\frac{3r{b(\pi)+r}), where r is the } number of prefix reversals applied by optimal algorithms. Finally, we present experimental results to support our analysis .