  The reliability redundancy allocation problem (RRAP) is a well-known tool in
system design, development, and management. The RRAP is always modeled as a
nonlinear mixed-integer non-deterministic polynomial-time hardness (NP-hard)
problem. To maximize the system reliability, the integer (component active
redundancy level) and real variables (component reliability) must be determined
to ensure that the cost limit and some nonlinear constraints are satisfied. In
this study, a bi-objective RRAP is formulated by changing the cost constraint
as a new goal, because it is necessary to balance the reliability and cost
impact for the entire system in practical applications. To solve the proposed
problem, a new simplified swarm optimization (SSO) with a penalty function, a
real one-type solution structure, a number-based self-adaptive new update
mechanism, a constrained nondominated-solution selection, and a new pBest
replacement policy is developed in terms of these structures selected from
full-factorial design to find the Pareto solutions efficiently and effectively.
The proposed SSO outperforms several metaheuristic state-of-the-art algorithms,
e.g., nondominated sorting genetic algorithm II (NSGA-II) and multi-objective
particle swarm optimization (MOPSO), according to experimental results for four
benchmark problems involving the bi-objective active RRAP.
