## 3 Mathematical Model Development
### Inputs
#### 3.2.2 Manufacturing Information  
Three kinds of feature are considered as follows.  
General features have two common machining constraints. One is \"First roughing and then finishing\", which means the precedence among the operations inside each feature is given. The other one is \"First face milling and then drilling\", which means that the drilling of the hole has to be completed after the machining of the one which the hole opens. For example, general features in the proposed case study can be classified into groups of features by the surfaces \(\text{FGS}=\{\text{FG}_{1},\dots,\text{FG}_{\text{G}},\dots,\text{FG}_{\text{ SN}}\}\), where \(\text{i}=1\dots\text{SN}\), and SN is the number of surfaces. For each \(\text{FG}_{i},\ i=1\dots\text{SN}\), there are \(\text{FNG}_{i}\) machining features with at least one operation that needs to be operated following its sequence. Operations in each \(\text{FG}_{i}\) can be described as:  
\[\text{OFG}_{i}=[A\ \times\text{FG}_{i}]=\begin{bmatrix}b_{1,1}&b_{1,2}&\cdots&b_{1, j}&\cdots&b_{1,FNGi}\\ b_{2,1}&b_{2,2}&&b_{2,j}&&b_{2,FNGi}\\ \vdots&\vdots&&\vdots&&\vdots\\ b_{k,1}&b_{k,2}&\cdots&b_{k,j}&\cdots&b_{k,FNGi}\\ \vdots&\vdots&&\vdots&&\vdots\\ b_{12,1}&b_{12,2}&\cdots&b_{12,j}&\cdots&b_{12,FNGi}\end{bmatrix} \tag{2}\]  
In the equation, \(\text{A}=\{\text{a}_{1},\text{a}_{2},\dots,\text{a}_{\text{m}}\}\) stands for the types of machining operations. In our example, \(\{\text{a}_{1},\dots,\text{a}_{3}\}\) are possible machining operations of a surface (rough milling, semi-finish milling, and finish milling); while \(\{\text{a}_{4},\dots,\text{a}_{12}\}\) are possible machining operations of a hole (drilling, gun drilling, core drilling, rough boring, semi-finish boring, finish boring, rough reaming, finish reaming, and tapping). Then, the element \(\text{b}_{\text{kj}}\) shows:  
\[b_{k,j}=\begin{cases}1&a_{k}\text{ have to be processed in feature }j\\ 0&a_{k}\text{ does not exist in feature }j\end{cases} \tag{3}\]  
The number of operations in each \(\text{OFG}_{i}\) is:  
\[\text{OPN}_{\text{OFG}_{i}}=\sum_{k=1}^{m}\sum_{j=1}^{\text{FNG}_{i}}b_{k,j}, \ i=1\dots\text{SN} \tag{4}\]  
Features for each datum system (\(\text{FDS}_{i}\), with \(\text{i}=1\dots\text{DSN}\)) are features that have to be completed at the same station where the datum system has never been used before. According to ISO GPS and ASME GD&T standards, a datum system is a reference system for dimensional and geometric tolerances, and in machining it consists of features that are usually considered in relation to clamping and positioning the parts. For example, the datum systems of the cylinder block in this paper usually have one surface and two corresponding pin holes. DSN is the total number of datum systems. The operations of a datum system are combined as a single datum operation. So DSN is also the number of operations.  
Special feature group (\(\text{SFG}_{i}\), \(\text{i}=1\dots\text{SGN}\)) is the set of features requiring special constraints, e.g., when operating two intersecting holes, both of them have to be machined at the same station and the longer hole should be machined before the shorter one; therefore, they are grouped into a special feature group, considered as a single special operation. SGN is the number of operations in the special feature groups.  
Eventually, the number of machining operations of a part can be calculated as:  
\[\text{OPN}=\sum_{i=1}^{\text{SN}}\text{OPN}_{\text{OFG}_{i}}+DSN+SGN \tag{5}\]  
For each operation \(\text{x}_{0}\) where \(\text{o}=1\)..OPN, its machining time \(\text{t}_{\text{m}}(\text{x}_{0})\) is given.