## 1 Introduction  
Modern products are becoming more and more complex, and their reliability level has an important influence on the development of the modern manufacturing industry. Thus, it is imperative to improve the product reliability optimization design [1]. It is an important analytical tool to enhance the inherent reliability of products, and the reasonable and effective reliability optimization design has a guiding significance for the later product detailed design [2; 3]. At the same time, the product cost is also a determining factor which can decide their market competitiveness, and, during application in practical engineering, the product optimal design scheme with high reliability and minimum cost is often desired by designers and customers [4; 5]. The reliability-based and cost-oriented product optimization has been a hot area of research for several years [6; 7]. The main common problems for the reliability optimization design of sophisticated mechatronic products are unquantifiable relationships among subsystems, modules, and components, as well as uncertain and insufficient product data. This results in the infeasibility of the methods which rely purely on mathematical algorithms. Although many efforts have been conducted on it, the efficient and effective method of reliability optimization design is still a pressing need because the conventional methods have some shortcomings that can not be ignored. Firstly, most conventional methods fail to process the uncertain information with effect in the product optimization design [8]. Uncertainty is a commonly encountered case in the initial stage of design because of the difficulties in collecting accurate data, due to, for example, the limitations of cost, time and resource [9; 10]. Therefore, reasonable reliability optimization design methodologies must have the ability to deal with such uncertainties [11; 12]. Bayesian theory and fuzzy mathematics were the initiate instruments to handle the uncertainties in conventional reliability optimization design [13; 14]. [PERSON] et al. [15] put forward the multi-attribute group decision-making (MAGDM) with the integration of techniques for order preference by similarity to ideal solution (TOPSIS) and interval-valued intuitionistic fuzzy sets, which can consider the reliability of information. [PERSON] et al. [16] put forward to adopt Bayesian model to capture cognitive causal relation in reliability investigation, and it successfully provided a potential tool for assessing error distribution. However, the Bayesian theory depends on a subjective judgment which brings inaccurate product optimization design plans, and fuzzy mathematics needs to make their membership functions be known in advance which is almost impossible for designers. To remedy these problems, the reliability optimization design with interval analysis was proposed where typical product variable configuration is represented using interval mathematical theory [17]. Interval analysis has recently been jointed with uncertainty algorithms, probability statistics, fuzzy mathematics, and grey system theory to get the reliability scheme with real rationality. However, these methods lack the consideration of the product failure which is linked closely with the reliability [18]. Critical product failures can bring heavy loss to economic value and society benefit, which means the analysis of product failure is essential for reliability optimization design [19; 20; 21]. Quantitative indicators of product failure are traditionally the failure frequency and failure severity, and some scholars dealt with the product reliability optimal design by the introduction of the qualitative analysis and quantitative calculation of failure mode effects analysis (FMEA) [22]. FMEA is used to describe product failure modes and their effects.