## 1 Introduction  
The reliability enhancement has increasingly become important in most systems because the system reliability is an important measure of system performance in many practical systems. For example, medical systems [1], video systems [2], traffic systems [3], industrial intelligence systems [4], network systems [5; 6; 7; 8; 9], wireless sensor networks [10; 11], power transmission networks [12], social networks [13], and Internet of Things [14; 15]. The evaluation and enhancement of system reliability must be optimized in the early design stage because the enhancement of reliability is the most efficient at this stage [16; 17]. With the objective of optimizing system reliability in various systems, the redundancy allocation problems (RAP), which has to only decide the redundancy of components, and the reliability redundancy allocation problems (RRAP), which has to simultaneously decide reliability and redundancy of components, are two well-known methods to improve the system reliability. RRAP that has been studied in this work is more complicated and difficult than RAP because it must determine the reliability and redundancy of components in the systems at the same time [18; 19; 20]. The researches of RRAP over the years have mainly used four network systems as benchmarks including series system, the network with series and parallel elements, a complex (bridge) system, and the overspeed protection of a gas turbine system, which are thus adopted in this study, as shown in following Figures 1-4. [PERSON] et al. researched the bi-objective RRAP including optimizing the system reliability and cost [18]. The cold-standby strategy in RRAP was adopted by several researchers such [PERSON], [PERSON] and [PERSON] [PERSON] and [PERSON] [19; 21; 22]. [PERSON] et al. considered the heterogeneous components in RRAP [23]. The heterogeneous components and mixed strategy including active and cold-standby strategy are studied by [PERSON] et al. [24]. [PERSON] shown the hybrid swarm algorithms in RRAP [25]. And [PERSON] presented a penalty guide in RRAP [26]. The fuzzy number that was originally invented by [PERSON] in 1965 [28] has been shown to be effective for solving the uncertainty condition, thus, this work adopts the fuzzy number for the uncertainties of systems components in the constraints of RRAP including cost, weight, and volume. However, the fuzzy numbers that applied to the constraints of RRAP including cost, weight, and volume finally must be transferred to a crisp value. [PERSON] and [PERSON] in 1998 first extended the chance-constrained programming, which let the constraints meet a certain probability value of confidence level such as \(p\) and was first developed by Charnes & Cooper in 1959 [29], to the fuzzy environment called the fuzzy chance-constrained programming [30]. After that, [PERSON] in 2004 developed the credibility theory [31] based on the fuzzy chance-constrained programming that has been successfully approved its effectiveness for transferring the fuzzy constraints to crisp constraints at a confidence level in several fields such as budget planning [32], and RAP [33]. So far, there are no scholars apply the fuzzy number and credibility theory to the comprehensive consideration of uncertainty in the constraints of RRAP. Therefore, this study adopts the fuzzy theory and credibility theory to solve the components uncertainty in the constraints of RRAP including cost, weight, and volume. The RRAP has been shown to be a NP-hard problem [18], thus, numerous metaheuristic algorithms are adopted to solve various RRAP researches such as genetic algorithm (GA) [21; 23; 27], particle swarm optimization (PSO) [22; 24], stochastic fractal search (SFS) [21], cuckoo search (CS) [25; 26], artificial bee colony algorithm (ABC) [34], and simplified swarm optimization (SSO) [18; 19; 20; 24].