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.
After revision:  Here we achieve an improved approximation ratio of (3-\frac{3r{b(\pi)+r}), where r is the } show quantitatively how approximation ratios of our algorithms improve with the increase of 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.