# Integrated Tolerance and Fixture Layout Design for Compliant Sheet Metal Assemblies
## 2 Problem Description
### Design Parameters
#### 2.1.2 Design Parameters of Fixture Layouts  
A fixture layout is defined by the location and the number of its locators on each part of the assembly. It is assumed that each part has a \(N-2-1\), \((N\geq 3)\) locating layout. Therefore, each part has one hole and one slot, and the optimization algorithm defines the optimal number of clamps (\(N\)). Hence, the design parameters of the fixture layout are the number of additional clamps, the location of holes, slots, and clamps, and the slot directions. A clamping unit can be positioned where a hole or slot is positioned. The hole or slot is clamped in this situation.  
The parameters to define the location of the locators can be either the coordinates of them or a node number. The later choice reduces the optimization size and complexities of the problem [37; 38]. Hence, the node number is used as the parameter to define the optimal location of each locator. Because this approach was developed by [PERSON] et al. [38] to define the design parameters of fixture layouts, the same parameters for the fixture layout  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(j=1\) & \(j=2\) & \(j=3\) &... & \(j=T_{i}\) \\ \hline \(i=1\) & \(t_{11}\) & \(t_{12}\) & \(t_{13}\) &... & \(t_{1T_{1}}\) \\ \(i=2\) & \(t_{21}\) & \(t_{22}\) & \(t_{23}\) &... & \(t_{2T_{2}}\) \\ \(i=3\) & \(t_{31}\) & \(t_{32}\) & \(t_{33}\) &... & \(t_{3T_{3}}\) \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) \\ \(i=m\) & \(t_{m1}\) & \(t_{m2}\) & \(t_{m3}\) &... & \(t_{mT_{m}}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The tolerance \(t_{ij}\) resulting from manufacturing part \(i\) through production strategy \(j\).  
are employed in this paper. These parameters are listed in Table 2, and they are introduced in this section.  
Not all areas of a part can be the location of a locator. Areas that are close to edges, bends, and areas that are not practically accessible cannot be the position of a locator. Therefore, the node numbers that can be utilized to be the position of a locator can be filtered, and the rest can be removed from the design space. Moreover, if the size of elements in each part is significantly smaller than the size of the part, changing the position of a locator from a node to the next node may not affect the results significantly. Therefore, to simplify the optimization problem, the feasible areas of each part can be divided into some smaller areas, and one node from each area (a node that is approximately in the center of the area) is considered to be the representative node of that area [38].  
After discretizing the feasible areas and selecting the candidate node, each node of part \(i\) is represented by \(k\) (\(k\in\mathbb{Z}^{+}\), \(1\leq k\leq n\)), where \(n\) is the number of all candidate nodes in part \(i\). The parameter \(\alpha_{ik}\) is introduced as a parameter to indicate whether the node \(k\) in part \(i\) is clamped. Accordingly, \(\alpha_{ik}=1\) indicates the node \(k\) of part \(i\) is clamped and \(\alpha_{ik}=0\) demonstrates that the node is not clamped.  
Each part has one hole and one slot. Therefore, two parameters are utilized to indicate the location (the node number) of the hole and the slot in each part. These parameters are \(\beta_{i,1}\), and \(\beta_{i,2}\) ( \((\beta_{i,1},\beta_{i,2}\in k)\)) for the hole and slot, respectively. \(\theta_{i}\) (\(0\leq\theta_{i}<180\)) is also introduced to define the slot direction in part \(i\). This parameter indicates the angle between the in-plane locking direction of the slot in part \(i\) and a fix arbitrary direction in the same plane.