A few cents is not much. "According to Ellis, when two notes are played together, a difference of 2 cents is noticeable, and a difference of 5 cents is heard as out of tune."
It could well be cheap strings, but it might not be.
Have you checked the intonation? I check after every change of strings.
The 12th fret harmonic and fretted note at 12th fret should all be in tune.
If it's still noticeably out of tune near the nut, then the nut probably is a bit high and needs careful re-cutting.
Virtually no instrument will be spot on for every fret, everywhere.