Puzzle n°16 (**) - The 15-Ring Chain

My cats love to play with anything that moves (or that I move...)
For instance, a chain of metal rings !

The other day, in the attic, I found 5 pieces of chain, each one made out of 3 rings like the one on the picture.

I would like to transform them into a single chain of 15 rings.
So it will be easier for me to dig it back from under the furniture where my cats love to hide their toys.

However, I am trying to make as few weldings as possible.
If I count as one the operation consisting in cutting open a ring and welding it close again,
what is the minimum number of operations needed for me to rebuild a 15-ring chain out of these 5 fragments ?

... I'm giving up ...