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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.