## 3 General-Configuration Aircraft Optimization Approach
### Stochastic Optimization Model for GCA
\begin{table}
\begin{tabular}{l l} \hline
**Indices** & **Representation** \\ \hline \(i\) & Index of product option groups. \\ \(I\) & Number of configurations in the product. \\ \(m_{i}\) & Index of options in the option group \(i\). \\ \(M_{i}\) & Number of options associated with the option group \(i\). \\ \(v_{i}\) & The least number of options should be chosen for the option group \(i\). \\ \(D_{i}\) & Disassemble time of options for the option group \(i\). \\ \(A_{i}\) & Re-assemble time of options for the option group \(i\). \\ \(j\) & Index of customers. \\ \(J\) & Number of customers. \\ \(\delta_{jim_{i}}\) & Binary random variable, equal to 1 if option \(m_{i}\) is chosen for the option group \(i\) by the customer \(j\). \\ \(\mathbf{\hat{o}}\) & Random matrix with three dimensions (customers, option groups, and options), a combination of orders. \\ \(k\) & Index of stations. \\ \(K\) & Number of stations in the production line. \\ \(T_{im,k}\) & Duration time of the configuration \(i\) performed as the option \(m_{i}\) \\ \(W_{k}\) & Maximal duration times reserved for options of option groups \\  & at the station \(k\). \\ \(x_{im_{i}}\) & Binary variable equals 1 if the option \(m_{i}\) is chosen for \\  & the option group \(i\). \\ \hline \end{tabular}
\end{table} 
Table 2: Notation. In addition to the objective function, the stochastic optimization model for GCA also consists of constraints, which are briefly described in the following part. Each option group has to choose at least the corresponding number of options:  
\[\sum_{m_{i}}x_{im_{i}}\geq v_{i},\,1\leq i\leq I \tag{5}\]  
The coupling relationships between the options of the option groups are divided into three types of constraints: assembly constraints, supplier constraints, and customer-oriented constraints. The assembly constraints originate from the assembly order of the options or option groups. Supplier constraints mainly refer to preferences toward options from the same supplier. The insufficient supply capacity of a supplier is also considered a supplier constraint. Customer-oriented constraints refer to factors such as product characteristics, customer preferences, and adaptability. The above three types of constraints are summarized and represented by two forbidden sets, an option coupling set \(F^{O}\), and an option group coupling set \(F^{G}\). Correspondingly, coupling constraints are described as follows:  
\[x_{im_{i}}+x_{jm_{i}}\leq 1,\,\forall(i,m_{i},j,m_{j})\in F^{O} \tag{6}\] \[\sum_{m_{i}}x_{im_{i}}+\sum_{m_{j}}x_{jm_{j}}\leq 1,\,\forall(i,j) \in F^{G} \tag{7}\]  
The Aircraft Moving Assembly Line (AMAL) is a new production model that replaces the traditional fixed assembly method in the aircraft industry. There are multiple work stations in the AMAL. The particular choice of option groups is assembled in one or several work stations. Considering the cycle time of the AMAL, the workforce of each station is limited. The workforce constraints in each station can be described as follows:  
\[\sum_{i}\sum_{m_{i}}T_{im_{i}k}x_{im_{i}}\leq C_{k},\,1\leq k\leq K \tag{8}\]  
Moreover, decision variables that represent the option choices should have binary values:  
\[x_{im_{i}}\in\{0,1\},\,1\leq i\leq I,\,1\leq m_{i}\leq M_{i} \tag{9}\]