Shortest number of editing move between two words I am looking for a data structure and an algorithm to compute the minimum number of changes required to transform one word into another, given the two words as inputs, where the only allowed changes are


add a letter at one of the extremities (for example, AB -> ABC),
duplicate and concatenate the whole word (for example, ABC -> ABCABC),
cut a word in two (the dual of the duplication move, ABCABC -> ABC + ABC),
delete one of the letters (for example, ABC -> AC), and
repeat one of the letters (for example, ABC -> ABBC).


For example, a minimal sequence of moves from ABC to BCBC is ABC -> BC (delete A) -> BCBC (duplication).

I don't have a background in computer science. Perhaps this is a well-known problem, but my Google search gave me nothing.

Do you know some related, well-defined problem?

Edit: As suggested in the answer by Anthony Labarre, I read some papers about the poset permutation/arrangement problem which is similar to the problem described above. Does anyone know more about this problem? Is this relevant?
I don't know whether this exact problem has been studied, but  Chaudhuri et al. studied the related tandem duplication-random loss problem: you are given a permutation, and you want to transform it into the identity permutation by 
 (1)  duplicating a segment of any length and appending the copy right after the original, then (2) deleting elements so that you obtain a new permutation instead of a string. Note that applying (1) then (2) accounts for one operation. 

Different variants can be defined according to the weight given to each operation, which in their paper depends on the width of the duplicated segments. 
They also study a similar problem with the whole genome duplication, which is exactly the kind of duplication you allow. I don't remember reading about work on this problem in the context of strings, but I hope this can at least give you a starting point for your searches.