This study introduces a time-dependent fuzzy multi-objective reliability redundancy allocation problem (TF-MORRAP) for the n-stage (level) series–parallel system. System reliability maximization and system cost minimization according to time by optimizing the redundant components counting at every stage of the system is the main objective of this study. This optimization is done by satisfying the entropy constraints with limited redundant components at every stage and in the whole system. The reliability and cost of every component are represented as triangular fuzzy numbers (TFN) to handle the uncertainty of input information of the system. According to time, the component reliability and cost decrease by some factor of their previous existing value. This factor follows the change in the length of radius of the inverse logarithmic spiral with respect to angle which is regarded as time here. The proposed problem is analyzed by using an over-speed protection system of a gas turbine. We compare the membership values of optimal solutions obtained by using two well-known techniques namely non-dominated sorting genetic algorithm-II (NSGA-II) and a multi-objective particle swarm optimization algorithm called NF-MOPSO. Various performances of the algorithms are compared to solve the aforementioned problem by using some performance metrics. NF-MOPSO shows the high satisfaction level of objective functions and better performance than NSGA-II.