# 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  
SysML lacks direct elements to represent variables. In multidisciplinary systems, a design variable refers to quantities to be optimized in the design process, which are independent of each other and unaffected by other variables. Design variables can be categorized based on their scope of influence into system design variables and local design variables. System design variables, also known as shared deterministic variables, have an effect across the entire system, whereas local design variables only impact a specific discipline. Local design variables are sometimes referred to as disciplinary variables or subspace design variables. Design variables can also be classified based on uncertainty into four types: deterministic design variables, aleatory uncertainty (AU) variables, epistemic uncertainty (EU) variables, and time-variant uncertainty variables. Additionally, variables can be classified as either discrete variables (DVs) or continuous variables (CVs) based on their value continuity. Since system design variables and local design variables can be defined and explored within optimization solvers, there is no need to define them separately in SysML system design models. Therefore, this paper focuses primarily on defining design variables based on their uncertainty. Another special type of variable in multidisciplinary systems is the coupled state variable (CSV). This refers to state variables from other disciplines that are used as inputs for the current discipline being analyzed. Since state variables can be represented by parameters, there is no need to extend the metamodel for them. Based on the above analysis, it is necessary to enhance the \"parameter\" type in SysML with additional attributes to define variable types suitable for multidisciplinary system optimization. This can be achieved by adding Tagged Values to specify the type of optimization variable a module attribute belongs to, as illustrated in Figure 3.  
To facilitate the definition of design variables in the SysML design model, as depicted in Figure 4, the \"Variable\" class extends the SysML metaclass \"Block\". Given that coupled state variables typically have associated equations and can be expressed through a continuous variable and a constraint block, no additional extensions are nee  
Figure 3: Variable type expansion in the value property. optimization variable metamodel \"Variable\" is instantiated into four types: continuous variables, discrete variables, aleatory variables, and epistemic variables. As shown in Figure 4, for the continuous variable configuration shapes in the SysML design model, the minimum and maximum values must be specified for their conformations to limit the range of the solver exploration space. For aleatory variables, at present, the use of probability theory to quantify aleatory uncertainty has been widely recognized in academia and engineering circles and has achieved great success. For aleatory uncertainties subject to other distributions, the most appropriate probability distribution (such as log-normal distribution, Weibull distribution, Gumbel distribution, etc.) should be selected for characterization, and the Rosenblatt method should be used to convert it to standard normal space. The conversion relationship of different probability distributions is shown in Table 2.  
For the characterization of aleatory variables, it is necessary to specify their means and variances, as well as their distribution types. As shown in Table 2, this paper defines the distribution type of aleatory variables as integer types from 1 to 5. When extracting the optimization model, its distribution type is determined by judgment. When the distribution type is 1 to 5, its distribution is defined as a normal distribution, a logarithmic distribution, a Weibull distribution, a Gumbel distribution, or a uniform distribution. Regarding cognitive variables, they can be classified into fuzzy uncertainty and interval uncertainty according to their quantification types, etc.