If we do a best-first search where the priority queue is ordered by h(s) + d(s), the search algorithm is called "A*" ("A-Star").

It can be proven that if h(s) never overestimates a state's distance to the final state, then A* search is optimal, that is, it will find a shortest solution if one exists.

As mentioned, the number of tiles out of place heuristic for the 8-Puzzle problem does not overestimate.