  In this paper we study several variations of the \emph{pancake flipping
problem}, which is also well known as the problem of \emph{sorting by prefix
reversals}. We consider the variations in the sorting process by adding with
prefix reversals other similar operations such as prefix transpositions and
prefix transreversals. These type of sorting problems have applications in
interconnection networks and computational biology. We first study the problem
of sorting unsigned permutations by prefix reversals and prefix transpositions
and present a 3-approximation algorithm for this problem. Then we give a
2-approximation algorithm for sorting by prefix reversals and prefix
transreversals. We also provide a 3-approximation algorithm for sorting by
prefix reversals and prefix transpositions where the operations are always
applied at the unsorted suffix of the permutation. We further analyze the
problem in more practical way and show quantitatively how approximation ratios
of our algorithms improve with the increase of number of prefix reversals
applied by optimal algorithms. Finally, we present experimental results to
support our analysis.
