By　using　the　outer　space　branch-and-reduction　scheme,　we　present　a　novel　algorithm　for　globally　optimizing　the　sum　of　several　affine　fractional　functions　problem　(SAFFP)　over　a　nonempty　compact　set.　For　providing　the　reliable　lower　bounds　in　the　searching　process　of　iterations,　we　devise　a　novel　linearizing　method　to　establish　the　affine　relaxation　problem　(ARP)　for　the　SAFFP.　Thus,　the　main　computational　work　involves　solving　a　series　of　ARP.　For　improving　the　convergence　speed　of　the　algorithm,　an　outer　space　region　reduction　technique　is　proposed　by　utilizing　the　objective　function　characteristics.　Through　computational　complexity　analysis,　we　estimate　the　algorithmic　maximum　iteration　times.　Finally,　numerical　comparison　results　are　given　to　reveal　the　algorithmic　computational　advantages.　©　2023　the　Author(s),　licensee　AIMS　Press.