The series-parallel (active) redundancy allocation problem with mixed components (RAP) involves setting reliable objectives for components or subsystems to meet the resource consumption constraint, e.g., the total cost. RAP has been an active research area for the past four decades. The NP-hard difficulties confronted by RAP are maintaining feasibility with respect to two constraints: cost and weight. A novel algorithm called the bound-rule-BAT (BRB) based on the binary-addition-tree algorithm (BAT), the dominance rule, and dynamic bounds are proposed to solve the exact solutions of the most famous RAP benchmark problems called the (33-variation) Fyffe RAP. From the experiments, the proposed BRB can solve the Fyffe RAP correctly under the assumption that the maximal number of components of each subsystem is eight, and this is the first exact-solution algorithm that can solve the Fyffe RAP within 8 seconds and 60 seconds if no reliability lower bound is used.