# Dimensional Tolerances in Mechanical Assemblies: A Cost-Based Optimization Approach
## 3 A Proposal for Tolerance Synthesis  
An analysis of the existing literature indicates that problems involving multiple chains are often tackled using heuristic methods because of their unique characteristics and complexity. The objective of the proposed optimization approach is to provide a practical and user-friendly tool. By adopting a heuristic approach instead of a complex mathematical treatment and by offering flexibility in selecting different cost functions, the proposed method is designed to be simple and adaptable for practical applications. The main objective of the cost-quality optimization task is to determine a set of tolerance values, denoted as \(t_{i}\), for \(n\) components. These tolerance values are synthesized in such a way that they minimize the total cost, denoted as \(CT\), which consists of both the manufacturing cost, denoted as \(C(t_{i})\), and the non-quality cost, denoted as \(Cq(t_{i})\). The proposed approach aims to find the optimal tolerance values that result in the lowest possible total cost \(CT\) while considering both manufacturing cost and non-quality cost factors  
\[CT=\sum_{i=1}^{n}C(t_{i})+\sum_{i=1}^{n}Cq(t_{i}). \tag{3}\]  
The graphical representation of (3) is depicted in Figure 4. In the synthesis process, it is essential to ensure that the assembly tolerance \(T\) satisfies one or more functional constraints denoted as \(R_{f}\), as  
\[T\leq R_{f}. \tag{4}\]  
These functional constraints define the acceptable limits or requirements that the assembly must meet in order to ensure proper functionality. The proposed pragmatic approach highlights two features that are not simultaneously addressed in the referenced works. Considering both features together, the optimization framework offers a comprehensive solution that takes into account their mutual influence and provides a more holistic approach to the problem. 1. The non-quality cost is an inherent aspect of the optimization process, influenced by the system configuration, as depicted in Figure 4. Therefore, the inclusion of an  
Figure 4: Graphical representation of the sum of manufacturing costs and non-quality costs. For each tolerance value \((t_{i})\), the total cost \(CT\) is the sum of the manufacturing cost \(C(t_{i})\) and the non-quality cost \(Cq(t_{i})\). The minimum value is the optimum total cost to be considered. economic safety factor, as indicated in (1) and (2), serves as an additional and significant control mechanism. By considering the economic safety factor, the optimization approach accounts for the potential financial implications of non-quality costs and ensures that the resulting solution aligns with cost-effectiveness considerations. 2. The pursuit of a global minimum cost, as demonstrated in several cited works, does not offer the flexibility to impose restrictions on the minimum values of component tolerances. This limitation hinders the ability to choose suitable manufacturing processes at a specific production site. It is important to note that lower tolerance values correspond to lower non-quality costs. However, such narrow tolerances may result in economically unviable processes for the manufacturer. However, the selection of a manufacturing process typically allows maximum tolerance values to be accommodated. This flexibility ensures that the chosen process remains within the realm of economic feasibility while still meeting the required quality standards. The formulation of this optimization proposal was influenced by an in-depth analysis of the limitations present in the current available optimization methods. The primary goal of developing this method was to provide a competitive solution to the problem of dimensional tolerance specification while minimizing the limitations observed in similar approaches, particularly their inherent complexity. Furthermore, it is important to note that the total tolerance cost is just one component among many that contribute to the overall assembly cost.