# Multidisciplinary Reliability Design Optimization Modeling Based on SysML
## 3 Representation of Optimization Problems Using SysML
### Definition of Extension Types for Optimization Types
#### 3.1.1 Optimization Variable Metamodel
For the convenience of processing, interval uncertainty is used uniformly in this paper for characterization. Be \begin{table}
\begin{tabular}{c c c} \hline \hline
**Distribution** & **Parameters** & **Convert Relationships** \\ \hline Normal distribution & \(\mu=\textit{mean value}\), \(\sigma\) & \(X=\mu+\sigma U\) \\  & & \(=\textit{standard deviation}\) & \(X=\mu+\sigma U\) \\ Log-normal distribution & \(\overline{\sigma}^{2}=\ln\left[1+\left(\frac{\sigma}{\mu}\right)^{2}\right]\) & \(X=\epsilon(\overline{\mu}+\overline{\sigma}U)\) \\  & \(\overline{\mu}=\ln(\mu)-0.5\sigma^{2}\) & \\ Weibull distribution & \(k>0\), \(\mu=\Lambda\Gamma\left(1+\frac{1}{k}\right)\) & \(X=\lambda\big{[}-\ln\left(\Phi(-U)^{*}\right)\big{]}^{\frac{1}{2}}\) \\ Gumbel distribution & \(\mu=\
u+\frac{0.577}{\kappa}\), \(\sigma=\frac{\overline{\mu}}{\sqrt{6\alpha}}\) & \(X=\
u-\frac{1}{\kappa}\ln[-\ln(\Phi(U))]\) \\ Uniform distribution & \(\mu=\frac{\mu+\frac{1}{2}}{\sigma}\), \(\sigma=\frac{b-a}{\sqrt{12}}\) & \(X=a+(b-a)\Phi(U)\) \\ \hline \hline \end{tabular} \({}^{*}\Phi(U)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{U}e^{-\frac{a}{\pi}}du\).  
\end{table} 
Table 2: Probability distributions and their x-u spatial transformation relationships. Figure 4: Optimization variable metamodel definitions (The \({}^{*}\) represents multiplicity, where 0..\({}^{*}\) indicates zero to an infinite number of instances, while 1..\({}^{*}\) signifies one or more instances. The same notation is used in Figures 5, 6, 7, 8, 11 and 12.) cause interval uncertainty requires fewer quantification parameters, only the range of its values needs to be determined, while other types of cognitive uncertainty can remove some information to become cognitive uncertainty. Therefore, in the SysML optimization definition, it is only necessary to define the value boundaries of cognitive variables. In the SysML design model, discrete variables often correspond to the selection of components, while the selection of components has two processes: the selection of a specific type of component and the selection of a specific number of components. The use of multiplicity and instances can be used here to characterize that, for a discrete variable, there is a class and a set of instances, \(\text{INS}=[\text{ins}_{1},\text{ins}_{2},\dots,\text{ins}_{m}]\), associated with it for characterizing the choice of the value taken by the variable. There are three situations:  
(1) The model element has only one instance decision. An instance, \(\text{ins}_{1}\), is identified by defining a bounded integer variable \(x\in[1\dots\ m]\);  
(2) The existence of a multiplicity of model element decisions. Identify the multiplicity of this model element by defining a bounded integer variable \(y\in[1\dots\ n]\);  
(3) Model elements have both instance decisions and multiplicity decisions. For this combination problem, create a Boolean value \(x_{ij}\), where  
\[x_{ij}=\left\{\begin{array}{l}1\ \text{ins}_{j}\ \text{for decision points with multiplicity}\ i\\ 0\ \The constraints are divided into equation constraints and inequality constraints, where the equation constraints need to determine the value of the state function in addition to the above definition, and the inequality constraints need to set the upper and lower boundaries of the state function. The state function is derived from the equation in the attribute. The extended relationship between the equation constraint and the inequality constraint is shown in Figure 6.