Global algorithm for addressing sum of linear ratios problem using the separable nature of relaxation problem

This paper proposed an algorithm based on the branch-and-bound framework for globally solving the sum of linear ratios problem (SLRP) with a large number of ratios and a small number of variables. First, we introduced new variables to construct an equivalent problem of the problem (SLRP). Then, using a new linear relaxation technique, we obtained the linear relaxation problem for the equivalent problem. By utilizing the separable nature of the linear relaxation problem, we computed the linear relaxation problem by solving its $p$ linear programming subproblems, thereby the lower bound for the problem (SLRP) could be obtained. Additionally, we conducted a theoretical analysis of the proposed algorithm and validated its feasibility and effectiveness through numerical experiments.