# Quantitative Model for the Value of the 3D Product Model Use in Production Processes
## 2 Materials and Methods  
In this study, we develop a quantitative value model for the deployment of the 3D product model in production and assembly processes. This model is based off prior research in which the authors found that the areas of value in implementing the 3D product model in the production and assembly process are as follows (listed in order of greatest to least qualitative value):  
1. Accuracy of assignment of the right parts, tools, work allocation, and work instructions;
2. Faster new product/model roll out to production;
3. Less time updating work instructions;
4. Quicker operator training;
5. Smoother transition for field use. Thus, the quantitative value model has been developed around these key areas. Given the way in which the 3D product model is used to qualitatively create value in each of these areas is discussed in prior research, this article will not focus on how the 3D product model is used in those applications but rather focus on the determination of the quantitative value of deploying the 3D product model in these five value areas. The remainder of this section proposes a quantitative model for determining the value of 3D product model use in production and assembly processes. The accuracy of assignment of the right parts, tools, work allocation, and work instructions is the area that industry respondents indicated they found the most value in implementing the 3D product model in assembly processes, at a level over 50% greater than any other value area [1]. When the right parts, tools, work allocation, and work instructions are assigned to the right production line stations, it enables employees to reduce variation in the production time per station. Reduction in variability of production time per station leads to the capability to reduce the takt time, and thus generates a greater throughput on the production line. Assuming a normal distribution in takt time variation by station, the model demonstrating this value is illustrated as follows:  
\(M=\) Number of production/assembly lines in a manufacturing plant  
\(m=\) A given production line in a manufacturing plant (1-\(M\))  
\(N_{m}=\) Number of stations on production line \(m\)  
\(n=\) A given station on production line \(m\) (1-\(N\))  
\(O_{m,n}=\) Number of operators at station \(n\) on production line \(o=\) A given operator at station \(n\) on production line \(m\) (1-\(O\))  
\(T_{m,n}=\) Target takt time of station \(n\) on production line \(m\)  
\(\sigma_{T_{m,n}}=\) Standard deviation of takt time at station \(n\) on production line \(m\)  
\(\beta_{m,n}=\) Probability of \(\hat{T}_{m,n}\leq T_{m}\)  
\(Z_{\beta_{m,n}}=Z\) value of \(\beta_{m,n}\)  
\(\hat{T}_{m,n}=\) Actual takt time of station \(n\) on production line \(m\) for a given production instance  
\[\hat{T}_{m,n}=T_{m,n}+Z_{\beta_{m,n}}\times\sigma_{T_{m,n}} \tag{1}\]  
\(T_{m}=\) Takt time of production line \(m\)  
\[T_{m}=\max_{n=1}^{N_{m}}\Bigl{\{}Z_{\beta_{m,n}}\times\sigma_{T_{m,n}}+T_{m,n} \Bigr{\}} \tag{2}\]  
Thus, if the Target takt time of station \(n\) on production line \(m\) (\(T_{m,n}\)) and the desired probability of the actual takt time of station \(n\) on production line \(m\) being less than or equal to the takt time of production line \(m\) (\(\beta_{m,n}\)) remain the same across all stations on the production line \(m\), then if the standard deviation can be decreased for each station \(n\) on line \(m\) (\(\sigma_{T_{m,n}}\)) the takt time of production line \(m\) can also be decreased (\(T_{m}\)). Note that this reduction in takt time of the production line \(m\) translates to a reduction in takt time of entire production process if the given production line is the bottleneck of the full manufacturing facility, otherwise speeding up the production line will not generate additional throughput through the production facility.