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<p style='text-indent:20px;'>This paper presents an effective algorithm for globally solving the sum of linear ratios problem (SLRP), which has broad applications in government planning, finance and investment, cluster analysis, game theory and so on. In this paper, by using a new linearization technique, the linear relaxation problem of the equivalent problem is constructed. Next, based on the linear relaxation problem and the branch-and-bound framework, an effective branch-and-bound algorithm for globally solving the problem (SLRP) is proposed. By analyzing the computational complexity of the proposed algorithm, the maximum number of iterations of the algorithm is derived. Numerical experiments are reported to verify the effectiveness and feasibility of the proposed algorithm. Finally, two practical application problems from power transportation and production planning are solved to verify the feasibility of the algorithm.</p>

<abstract><p>This paper proposes an outer space branching search method, which is used to globally solve the generalized affine fractional optimization problem (GAFOP). First, we will convert the GAFOP into an equivalent problem (EP). Next, we structure the linear relaxation problem (LRP) of the EP by using the linearization technique. By subsequently partitioning the initial outer space rectangle and successively solving a series of LRPs, the proposed algorithm globally converges to the optimum solution of the GAFOP. Finally, comparisons of numerical results are reported to show the superiority and the effectiveness of the presented algorithm.</p></abstract>