# Multidisciplinary Reliability Design Optimization Modeling Based on SysML
## 2 Related Works  
### RBMDO  
In 1982, [PERSON] first proposed the concept of Multidisciplinary Design Optimization (MDO), laying the foundation for its development [10]. Subsequently, Reliability-Based Multidisciplinary Design Optimization (RBMDO), aimed at improving the design quality of complex engineering systems by considering the impact of uncertainties, was introduced [8]. RBMDO comprehensively takes into account various uncertainty factors in a system, reasonably assessing their impact on the output functions of the multidisciplinary system to ensure that the MDO design results meet specified reliability requirements. MDO has developed over nearly 30 years, yet there is no universally accepted definition of MDO internationally. Nonetheless, its characteristics can be summarized as follows: it involves a thorough exploration and utilization of interactions within the system; it requires the analysis of interactions between disciplines or subsystems and the use of these interactions for integrated system optimization [11]. Essentially, MDO is a comprehensive method for addressing complex engineering system design challenges. By exploring and leveraging the synergistic mechanisms of interactions within the engineering system and considering the interplay between disciplines, MDO aims to optimize the design of complex engineering systems from a systemic perspective, with goals of improving product performance, reducing costs, and shortening design cycles. MDO leverages the interactions between disciplines to achieve an overall optimal solution for the system. The traditional Deterministic Multidisciplinary Design Optimization (DMDO) mathematical model can be described as Equation (1) [8]:  
\[ \begin{array}{c}\underset{(DV=d_{s},d)}{\min}\ \ f(\mathbf{d}_{s},\mathbf{d},\mathbf{y})\\ s.t.\ \ g^{(i)}(\mathbf{d}_{s},\mathbf{d}_{i},\mathbf{y}_{i})\leq 0\\ h^{(i)}(\mathbf{d}_{s},\mathbf{d}_{i},\mathbf{y}_{i})=0\\ \mathbf{d}_{s}^{L}\leq\mathbf{d}_{s}\leq\mathbf{d}_{s}^{L|},\mathbf{d}_{i}^{L}\leq\mathbf{d}_{i} \leq\mathbf{d}_{i}^{L|}\\ i=1,2,\cdots,nd\end{array}  \tag{1}\]  
The deterministic design variables in Equation (1) are \(\mathbf{d}_{s}\) and \(\mathbf{d}\). \(\mathbf{d}_{s}\) is the shared deterministic design variable, \(\mathbf{d}=(\mathbf{d}_{i},\ i=1\sim nd)\) is the local design variable consisting of all disciplines, and \(\mathbf{d}_{i}\) is the deterministic local design variable of the discipline \(i\). The term \"shared design variables\" refers to design variables that are common to all disciplines, while \"local design variables\" refers to design variables of a particular discipline. \(\mathbf{d}_{s}^{L|}\) and \(\mathbf{d}_{s}^{L}\) are the upper and lower bounds of \(\mathbf{d}_{s}\), and \(\mathbf{d}_{i}^{L|}\) and \(\mathbf{d}_{i}^{L}\) are the upper and lower bounds of \(\mathbf{d}_{i}\). \(f(\bullet)\) are the optimization objectives of DMDO, \(g^{(i)}\) and \(h^{(i)}\) are inequality constraints and equation constraints in the discipline \(i,i=1\sim nd\), and \(nd\) is the total number of all disciplines included in the complex engineering system. \(y=\{y_{1\bullet},y_{2\bullet},\ldots,y_{md\bullet}\}\) is the coupled variable that discipline \(i\) outputs to other disciplines and is the set of all coupled state variables of the MDO optimization problem. \(y_{i\bullet}=\{y_{ij},j=1\sim nd,j\
eq i\}\), and \(y_{i\bullet}\) is the coupled state variable of the discipline \(i\) output to all other disciplines. Similarly, \(y_{i\bullet}\) is the coupled state variable of all other disciplines input to the discipline \(i\). Uncertainties commonly exist in all stages of engineering design, and as the complexity of products increases, uncertainties (including types and quantities) also increase.