# Dimensional Tolerances in Mechanical Assemblies: A Cost-Based Optimization Approach
## 3 A Proposal for Tolerance Synthesis
To assess its significance for a specific manufacturer, its magnitude should be evaluated in relation to other cost factors. By considering the relative importance of the total tolerance cost in the context of the entire cost structure, manufacturers can make informed decisions about the optimization of tolerances and their potential impact on the overall affordability of the product. The Proposed Method  
The following guidelines were considered:  
1. The definition of assembly functional constraints involves establishing quality criteria that consider both the alignment with the customer's requirements and the cost-effectiveness of the product's performance in terms of quality considerations. 2. The specification of tolerance ranges for assembly participant dimensions in the chain analysis is performed without imposing initial restrictions on maximum values. This process draws upon tables found in literature sources such as [PERSON] [44,45], or preferably, on a company's internal data. By adopting this approach, it becomes possible to minimize the initial cost associated with each component. 3. The chain tolerance analysis is conducted utilizing the Root Sum Square (RSS) method. 4. The tolerance synthesis involves optimizing the manufacturing costs of components based on their respective tolerance ranges. The algorithm is designed to identify the most effective distribution of tolerance ranges among the components, ensuring compliance with the assembly tolerance range specified by the adopted functional constraints. To ensure flexibility, the algorithm is capable of calculating both simple and interrelated chains. In cases where multiple chains are present, the calculation can be performed sequentially. The main algorithm starts by running the first (main)tolerance chain. Tolerances for components shared with other chains, which have already been optimized in the first chain, are considered predetermined and \"frozen\" in subsequent steps. This enables the main algorithm to re-run with only the newly determinable tolerances of the additional chains. If required, a preliminary sensitivity test can be performed to select the preferred main chain (without involving tolerance calculation). This entails running a preliminary analysis by treating the common components of interrelated chains as a single chain. This allows for evaluating the sensitivity of the components and helps in determining the main chain that will be considered in the subsequent algorithm run. This is an additional distinctive feature of the method. Although a discretized linear function was used due to its practical applicability in the industry, the algorithm and the MatLab(r) software do not impose any restrictions on the choice of cost-tolerance function. 5. The system provides the capability to specify both maximum (for cost optimization) and minimum (for process feasibility) values for determinable tolerances. This feature plays a crucial role in the selection and/or restriction of manufacturing processes during the design phase of each component. It enables the method to be employed by various manufacturers, accommodating their specific limitations and requirements. The optimization algorithm consists of two modules. The first module focuses on establishing functional constraints and is associated with Guideline (1). The chains are arranged in ascending order on the basis of the magnitude of their respective functional constraint limits. The second module of the algorithm is responsible for performing the optimization process. The step-by-step procedure is outlined in Algorithm 1. The algorithm assumes that an assembly consists of \(m\) parts and \(p\) chains. Initially, a set of tolerances \(\{t_{k}\}\) is provided as input. Furthermore, the minimum tolerance value for each part is specified based on the available manufacturing process, denoted as \(\{\min(t_{k})\}\). These inputs correspond to Guidelines (2) and (5), respectively.