# Dimensional Tolerances in Mechanical Assemblies: A Cost-Based Optimization Approach
## 3 A Proposal for Tolerance Synthesis
Consequently, the tolerances of the common items shared by both chains are defined to comply with the more restrictive chain, which is the secondary chain. These tolerances are then used to calculate the tolerances of the primary chain. It is important to note that the total cost is not simply the sum of costs \(C1\) and \(C2\), since common items are considered only once to avoid duplication. Figure 4.6

The mandatory functional constraint values for the two chains, \(Rf_{1}=0.79\) mm and \(Rf_{2}=0.63\) mm, based on the cost of quality loss, indicate that the input data \(Spec\) do not meet the following requirements:

1. class IT14 at chain #1;
2. both classes IT13 and IT14 at chain #2. The values corresponding to these functional constraints are highlighted in bold in Figure 4.7 The columns containing these values are duplicated in Table 2, which is used for the tolerance synthesis process. In each column, the subsequent rows in the table present the following results:

* The tolerances T1 and T2 of the primary and secondary chains, respectively, were calculated using the RSS method. The program's data entry routine incorporated (5) and (7). * The relative costs C1 and C2 of the chains were calculated using the program's cost routine, applying (6) and (8). * The relative total cost (CT) of the assembly, composed of the two chains, was calculated using (9). ### Tolerance Synthesis Process

In Table 2, the results of the optimization calculation for the IT14 class values and the additional specification of the example are presented in the fourth and sixth columns,

\begin{table}
\begin{tabular}{l l l l l l l l l l l l} \hline \hline \multicolumn{1}{c}{} & \multicolumn{1}{c}{**Classes IT for Bi-Directional Tolerances According to Means**} \\ \hline
**Item** & **Mean** & **IT6** & **IT7** & **IT8** & **IT9** & **IT10** & **IT11** & **IT12** & **IT13** & **IT14** & **Spec. ** \\ \hline
1 & 35. & 0.008 & 0.013 & 0.020 & 0.031 & 0.050 & 0.080 & 0.



respectively. IT14 is the roughest ISO class selected for the case. Consequently, any tolerance above it is preferable to the lowest cost attainable when starting the optimization process. Optimization will increase accuracy (increase in manufacturing cost) depending on the other requirements (variables) involved. Processes are defined according to the specified tolerances. Different machines used in the same process (for example, lathes for turning) can present different accuracy capabilities, mainly due to the robustness and precision of the machine. ### Discussion

The analysis of the data summarized in Tables 1 and 2 reveals the following findings:

1. The tolerances of the primary and secondary chains in the IT14 class and Spec. columns in Table 1 have the lowest relative costs compared to other classes. However, they do not satisfy the functional constraints due to the specified tolerances of the components. 2. The relative costs of the chains and the total cost of the non-optimized allocation processes, shown in the last columns of Table 2, are higher than those of the optimized processes. This demonstrates the effectiveness of the proposed method. 3. It was observed that the conventional allocation method, which uses equal tolerances (not shown), yields cost results that are close to the optimized values. However, upon analyzing the allocated values for the components, it was found that they may not comply with the process capability, especially when considering larger dimensions. In addition to optimizing manufacturing costs, the proposed method allows for a restriction that ensures flexibility in dealing with manufacturing feasibility. 4. The proposed method is characterized by its simplicity and quick response time (less than one minute of program run with an Intel CORE i5\({}^{\otimes}\) processor). This allows for iterative attempts at optimized values, with validation by process experts if necessary, until an optimal solution is adopted.