To determine the relationship between geometric deviation parameters and production costs, experience in the form of known values and subsequent interpolation can be used on the one hand, or an arbitrarily detailed modelling of the costs can be aimed for on the other hand, see section “Determining the Manufacturing Process Costs for Conventional Processes in Gearbox Production.” Subsequently, a quality can be assigned to the total costs on the basis of measured values, production simulations or empirical knowledge. Modelling approaches for cost-tolerance functions (C-T functions) were therefore developed with the aim of enabling general applicability. The generally available data basis is not extensive, which poses a challenge. Many investigations were carried out under specific boundary conditions, which makes transferability to other manufacturing processes, tools or quantities challenging (Ref. 5–7). If not only the individual process costs are considered, but the balance sheet limits are extended to the total costs of an assembly, additional costs arise due to fully assembled units. In addition to the individual component manufactured outside the tolerance, these include other good parts that are also discarded. If the individual tolerance is extended, the probability increases that the assembly will no longer fulfil its function. This approach results in the optimum for the total costs of an assembly for a simplified example shown in Figure 1, top left (Ref. 8).

For a generalized cost-tolerance description of the individual processes, He presents analytical approaches based on exponential functions and parameterizes these using a survey. He uses the developed correlations as a criterion for a generalized tolerance design. He compiles the costs for an acceptable workpiece property from individual cost items. The machine costs are only a subordinate part of this, see Figure 1, top right (Ref. 9). Existing cost-tolerance modelling covers a wide range in addition to the actual production, which was described by Andolfatto et al. with a generalized formula for the individual costs Ci with hyperbolic, exponential and linear components, see Figure 1, bottom left (Ref. 10). The C-T functions of a manufacturing process should fulfil three conditions, which are in Equations 1–3.

The relationship between a manufacturing process and the appropriate analytical description model is not always clear. Sanz-Lobera et al., therefore, present an alternative approach for determining a cost-tolerance relationship using measurement data and the assumed scatter shape of a geometric feature. He comes to the realization that the scatter shape has an important influence on the cost-tolerance function and presents equations for some distributions (Ref. 11). Hallmann et al. also described the selection of suitable cost-tolerance functions as challenging. In addition, the approach with higher-order methods, such as Artificial Neural or Fuzzy Networks, is mentioned there in order to identify suitable functional relationships. In particular, the fact that different processes can achieve identical qualities at different costs is also addressed there, see Figure 1, bottom right (Ref. 5).