Sum through dissimilar sheets

I am trying to sum a total sheet where the products vary and the site names getted added. The attached sample gives a getter idea.

Thanks for any ideas. I have used INDIRECT but becuase of the size of the sheet (up to 50 sites) it is far too slow. SUMIF falls over becuase the new site names are not known in advance and this mean having to re-write the SUMIF formula.