I have learned several heuristic functions which are admissible to deal with the classical 8 puzzle problem, and I know you can multiply a factor to a admissible function to make it inadmissible, however, I wonder is there any other inadmissible heuristic function for the 8 puzzle problem?
There are all sorts of inadmissible heuristics to this puzzle. An inadmissible heuristic just needs to overestimate the number of steps it will take to solve a given puzzle, and so one simple inadmissible heuristic would be
h(S) = infinity
Since any puzzle that's solvable can be solved in fewer than infinity steps, the heuristic is inadmissible.
A much trickier and more interesting question would be what good admissible heuristics are out there, since they require you to give the largest possible value you can that doesn't overestimate the distance. For that, I don't have a good answer. :-)