# Quantitative Model for the Value of the 3D Product Model Use in Production Processes
## 2 Materials and Methods
### Quicker Operator Training  
The fourth of five categories, as seen in the first paragraph of the Materials and Methods section, in which industry respondents indicated they created value by implementing the use of the 3D product model in the production process is in quicker operator training. By having the 3D product model use in the production process, employees that need to be trained on a new or updated production process can see the 3D model of the product and gain a visual representation for the objective to be achieved, increasing the speed at which they learn to perform the operation within the target takt time for station \(n\) on production line \(m\) (\(T_{m,n}\)). Here we apply the learning curve theory [25] to model the value that the 3D product model use in the production process creates via quicker operator training:  
\(\Psi_{m,m,n,0}=\) Time to produce the 1 st unit by operator \(o\) at station \(n\) on production line \(m\)  
\(L_{m,n,o}=\) Learning curve rate for operator \(o\) at station \(n\) on production line \(m\)  
\(\delta=\) Production unit number  
\(b=\) Learning curve factor  
\[b=\frac{\ln(L_{m,n,o})}{\ln(2)} \tag{17}\]  
\(\Psi_{m,n,o,\delta}=\) Amount of time to produce unit \(\delta\) by operator \(o\) at station \(n\) on production line \(m\)  
\[\Psi_{m,n,o,\delta}=\Psi_{m,n,o,1}\times\delta^{b} \tag{18}\]  
\(\Psi_{m,o}=\) Time operator \(o\) spends training at station \(n\) on production line \(m\)  
\[\Psi_{m,n,o}=\sum_{\delta=1}^{\Psi_{m,n,o,\delta}\leq\overline{\Gamma}_{m,n}} \Psi_{m,n,o,\delta} \tag{19}\]  
\(x_{m,n,o}=\) Number of required trainings per operator \(o\) at station \(n\) on production line \(m\) per year (due to turnover, minor work instruction/product model updates, etc.)  
\(X_{m,n}=\) Number of employees trainings at station \(n\) on production line \(m\) per year  
\[X_{m,n}=O_{m,n}\times E+\sum_{o=1}^{O_{m,n}}x_{m,n,o} \tag{20}\]  
\(\Psi_{m}=\) Total time spent training on station \(n\) on production line \(m\)  
\[\Psi_{m,n}=\sum_{o=1}^{X_{m,n}}\Psi_{m,n,o} \tag{21}\]\(\gamma_{m,n,\rho}=\) Hours worked by operator \(o\) at station \(n\) on production line \(m\)  
\(\zeta_{m,n,o}=\) Cost of labor per hour for operator \(o\) at station \(n\) on production line \(m\)  
\(\zeta_{m}=\) Average cost of labor per hour for operators at station \(n\) on production line \(m\)  
\[\overline{\zeta}_{m,n}=\left(\sum_{o=1}^{O_{m,n}}\zeta_{m,n,\rho}\times\gamma_ {m,n,o}\right)/\sum_{o=1}^{O_{m,n}}\gamma_{m,n,o} \tag{22}\]  
\(\Phi_{m}=\) Cost of training for production line \(m\)  
\[\Phi_{m}=\sum_{n=1}^{N_{m}}\overline{\zeta}_{m,n}\times\Psi_{m,n} \tag{23}\]  
\(\Phi_{Without3D,m}=\) Cost of training for production line \(m\) (\(\Phi_{m}\)) prior to 3D product model implementation in production processes  
\(\Phi_{With3D,m}=\) Cost of training for production line \(m\) (\(\Phi_{m}\)) after 3D product model implementation in production processes  
\(P_{\Phi_{m}}=\) Incremental profitability via reduction in time to update work instructions  
\[P_{\Phi_{m}}=\Phi_{Without3D,m}-\Phi_{With3D,m} \tag{24}\]