# Dimensional Tolerances in Mechanical Assemblies: A Cost-Based Optimization Approach
## 1 Introduction  
Manufacturing processes inherently involve uncertainties, making it impractical to achieve exact geometries and dimensions for every part. When defining the geometry and dimensions of product features, it becomes necessary to account for these uncertainties, specifically the manufacturing variation, while ensuring that the final result allows the product to fulfill its intended design function. To specify a range of acceptable variations, it is important to consider the sensitivity of the product function to geometric and dimensional uncertainties, although at a lower level of magnitude. These acceptable ranges of variations are commonly referred to as \"tolerances\", which are defined by maximum and minimum values. The concept of tolerance development is illustrated in Figure 1. The tolerance range, representing the specified extent of variation, is applicable to components, assemblies of components, or simply referred to as \"assemblies\". In practice, the specification of dimensional tolerances becomes relevant when mechanical components interact through contact surfaces, forming a mechanical assembly. When determining the tolerances for a product, achieving a balance among the following factors is a recurring concern:  
1. Functional design requirements, also known as design constraints, encompass both functional and quality considerations. These constraints include aspects such as fitting with other assemblies, alignment between shafts, lubrication and sealing requirements, flow and thermal considerations, as well as visual and aesthetic requirements, among others. Typically, these constraints are identified as functional key characteristics (FKCs) or simply key characteristics (KCs). The accuracy and precision of manufacturing processes play a crucial role in specifying tolerances. The dimensional variation in manufacturing processes is influenced by multiple factors, including the capability and condition of the machinery being used, the setup of the process, the fixture holding the part, vibrations, tool temperature, and environmental conditions. These factors collectively contribute to the overall variation in dimensions during manufacturing. 3. Manufacturing costs, which depend on the chosen processes, have a direct impact on the overall cost of the product. These manufacturing costs directly influence the retail price of the product and, subsequently, its competitiveness on the market. Therefore, careful consideration of manufacturing costs is essential to ensure a competitive position in the market. The advantages of tolerance-cost optimization are widely recognized. However, its full potential remains untapped due to the complexity and limited comprehensibility of existing methods found in the literature [1]. This study aims to address this gap by introducing a more flexible and simplified approach to tolerance-cost optimization. The primary objective is to achieve the optimal balance between manufacturing cost and product quality. To accomplish this, a novel cost-based optimization heuristic is proposed for the process of specifying tolerances in both individual parts and assemblies. In the optimization process, various other design factors are taken into account as variables. The algorithm developed takes into account both simplicity and flexibility, allowing for parameter adjustments that can be tailored to specific industrial facilities. This customization capability enables the algorithm to effectively incorporate specific and local cost precision data in the development of the models that it utilizes. Moreover, the proposed optimization process has the capability to handle multiple tolerance chains simultaneously, which makes it different from existing methods where this feature is often overlooked. This ability to consider multiple tolerance chains is a notable differentiating factor of the proposed approach. The structure of this paper is organized as follows.