## 3 Detailed Operators
### Adoption of Cultural-Based DMOPSO Using Crowding Distance Sorting  
This process corresponds to Stage VI in Figure 1, and is described in Section 2. Compared with conventional algorithms, particle swarm optimization (PSO) is a simple and effective random algorithm to solve constrained problems [37; 38]. As shown in Equation (13), the reliability-based and cost-oriented product optimization model has two optimization objectives, the maximum reliability and minimum cost, but they are negatively correlated. DMOPSO has received extensive attention by scholars in recent years [39]. The outstanding feature of DMOPSO is the combination of strong convergence ability and prominent diversity of Pareto solutions, but it still has some defects. On the one hand, the complexity of the diversity preservation for the Pareto set and global optimum update strategy brings a high computational cost. On the other hand, DMOPSO often falls into a local extreme because of its relatively scarce overall searching ability. To overcome these difficulties, cultured-based DMOPSO was proposed with the elitism strategy and crowding distance operator which brings the Pareto front closing to the accurate optimum in a more economical way [31]. Therefore, cultured-based DMOPSO using crowding distance sorting is adopted for the mathematical model of reliability-based and cost-oriented product optimization. [PERSON] et al. [31] elaborated the corresponding concrete implementing process and properties of this arithmetic.  
The Pareto front has been widely used in finding the Pareto-optimal solutions to double object optimization problems. The Pareto front concept is derived from economics and engineering. It is astraightforward way to explore the trade-offs in double objective optimization. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set is a set of choices that are Pareto efficient. The Pareto frontier is particularly useful in engineering by restricting attention to the set of choices that are Pareto-efficient. Designers can make tradeoffs within this set rather than considering the full range of every parameter.  
In this paper, the Pareto front is applied to the reliability-based and cost-oriented product optimization model proposed in Section 3.4, which also has two optimization objectives: the maximum reliability and minimum cost. This product optimization model has two kinds of parameters represented by \(w_{i}^{d}\) and \(w_{i}^{f}\) (\(i=1,\,2,\,\ldots,\,n\), and \(n\) is the total number of subsystems of the target product obtained by the subsystem division based on failure decoupling in Section 3.1). \(w_{i}^{d}\) and \(w_{i}^{f}\) represent the weight considering the development cost of the \(i\)th subsystem and the weight considering product failures of the \(i\)th subsystem, respectively. In fact, the values of \(w_{i}^{d}\) and \(w_{i}^{f}\) in the product optimization model are computed by fuzzy logic and interval mathematics at first. Specifically, \(w_{i}^{f}\) is calculated by the fuzzy logic of FRPN in Section 3.2, and \(w_{i}^{d}\) is calculated by the interval mathematics with the method in Section 3.3. In this way, the mathematical model of reliability-based and cost-oriented optimization model proposed in this paper can be regared as a traditional double objective optimization problem. Then, the Pareto front can be adopted to find the best solutions. In this way, the Pareto front presented in this paper is obtained with fuzzy logic. Therefore, the Pareto front in fuzzy logic is the same as the traditional Pareto front in essence.