# Integrated Tolerance and Fixture Layout Design for Compliant Sheet Metal Assemblies
## 2 Problem Description
### Objectives  
Part tolerances and fixture layouts directly influence the production costs and geometrical quality of the assemblies. Depending on the application, both or one of these parameters can be considered as the objective of the optimization. The minimal geometrical quality of the assembly is commonly defined through tolerances on the KPCs of the assembly. Thereby, the objective is defined as minimizing the cost of part production while fulfilling the geometrical requirements of the assemblies.  
Each production scenario is associated with a production cost. The cost is commonly higher for the scenarios in which the produced parts have a greater geometrical quality. Supposing that the scenario \(j\) for part \(i\) costs \(c_{ij}\), the total cost of production can be determined by Equation (4).  
\[C=\sum_{i=1}^{m}\sum_{j=1}^{T_{i}}c_{ij}x_{ij} \tag{4}\]  
The minimal geometrical requirement is a problem constraint. However, this problem can be added to the objective function as a penalty. The penalty function should increase the value of the objective function when the geometrical quality is lower than the required quality and it does not add anything when it is higher.  
The deviation of each part may differ from the others in KPCs during mass production. Therefore, a statistical measurement is commonly utilized in order to evaluate whether the minimal requirements in KPCs are satisfied. In cases where the deviations are normally  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Variable** & **Feasible Values** \\ \hline Production scenario of part \(i\) & \(x_{ij}\) & 0, 1 \\ Being clamped or not for the node \(k\) of part \(i\) & \(\alpha_{ik}\) & 0, 1 \\ Assigned node to the hole & \(\beta_{i,1}\) & 1, 2,..., \(n\). \\ Assigned node to the slot & \(\beta_{i,2}\) & 1, 2,..., \(n\). \\ Slot direction & \(\theta_{i}\) & [0 180] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Design parameters.  
distributed, the process capability index (\(C_{pk}\)) can be utilized to evaluate the geometrical quality. Equation (5) represents the definitions of this parameter.  
\[C_{pk}=\min\{\frac{UL-\bar{d}}{3\sigma},\frac{\bar{d}-LL}{3\sigma}\} \tag{5}\]  
In this equation, \(UL\) and \(LL\) represent the upper and lower limits of the KPC, respectively. The parameter \(\sigma\) is the standard deviation of the KPC and \(\bar{d}\) is the mean value of its deviations. If the KPCs are defined by geometrical tolerances (as they are in the sample case utilized in this paper), the magnitude of deviation is utilized for evaluation, which cannot be negative. Consequently, the distribution of the deviation is not normal, and Equation (5) cannot be utilized to evaluate it. A more general definition of capability index, which is referred to as [PERSON]'s method [47], is utilized to address this issue. Equation (6) represents this definition. In this equation, \(\zeta_{0.99865}\) and \(\zeta_{0.00135}\) are the upper and lower 0.135 th percentiles of the distribution, respectively, and \(\zeta_{0.5}\) is the median of the distribution instead of its mean. Because there is no lower limit in the magnitude of the deviation, \(C_{pu}\) is utilized in this paper to measure the geometrical quality.  
\[C_{pk}=\min\{\frac{UL-\zeta_{0.5}}{\zeta_{0.99865}-\zeta_{0.5}},\frac{\zeta_{0. 5}-LL}{\zeta_{0.5}-\zeta_{0.00135}}\}=\min\{C_{pu},C_{pl}\} \tag{6}\]  
A \(C_{pu}\geq 1.67\) indicates 99.99% of the produced assemblies satisfy the defined limits of KPCs (if the 0.001 th percentile is used in the formulation, \(C_{pu}\geq 1\) indicates the same quality). Hence, the required geometrical is defined as having \(C_{pu}\geq 1.67\) for each KPC. If this requirement is not satisfied, then a penalty relative to its value will be considered as the penalty. Equation (7) represents this penalty function.  
\[p(C_{pu})=\begin{cases}0&C_{pu}\geq 1.67\\ \frac{1.67}{C_{pu}}&C_{pu}<1.67\end{cases} \tag{7}\]