The pressing of materials depends on a large number of parameters. There are complex relationships among these parameters, as well as between these parameters and the material and the quality requirements. As already mentioned in Section 2, there is relatively little knowledge available about which parameter values achieve the desired result. Even for an expert it is nearly impossible to find exact adjustments at once. To find the dependencies between various parameters, the product engineer usually tries several possibilities. The results of these trials are represented in the Product Knowledge Base. In the Production Knowledge Base, we will thus represent the regularities which are (supposedly) valid for the production process, in general. More specifically, we are concerned with the different parameter values for manufacturing GMT products with a hydraulic press (see figure 1).
The results or such scientific experiments are most often summarized by a linear equation, that is obtained by a regression analysis or by an Analyis of Variance (cf. [\protect\citeauthoryearKrottmaier1991]. Such an equation may for instance take the form:
Although such numeric equations are quite useful and have a broad field of application in research and industrial practice, there are also a few disadvantages, which can be compensated by a more abstract and qualitative description. One problem lies in the fact, the each experiment yields a new equation and it may be quite difficult for any practitioner (and even researcher) to derive a set of general regularities from the various equations. Secondly, these equations hold only within certain limits.This is, however, not directly represented by the equation. For instance, increasing the pressing force beyond certain limits will not increase the surface area in the way, that is predicted by the linear equation, but may instead damage the press. In other words, there is an upper and lower bound on the parameters as well as on the values of the criterion variable (e.g. the surface area).
In addition to such numerical representations, we therefore propose a more abstract and qualitiative description for representing the general knowledge from the various cases. Unlike the numerical equation, we assume upper and lower bounds for the criterion variable, whose values are denoted qualitatively, like for instance by ``large'', ``medium'' or ``small''. In other words, there is for instance no value that is smaller than ``very small'' and no value that is larger than ``very large''. As a consequence of these bounds, the qualitative addition operation, which we denote by , can no longer be a closed operation. In order to embody these limitations, we define the qualitative addition operation in the following way. Let denote a set of qualitative descriptors, like an, which we could for instance also call =very small, =small, =medium, ...=very large. We postulate that the set is weakly ordered. Since the cartesian product contains all logically possible qualitative additions of the form , where and are in , those that can actually be formed must constitute a subset of . Thus, if is in , then and can be qualitatively added and so is in . This means that the operation is a function from into . In order to account for the fact, that not all qualitiative additions are possible, we define a qualitative structure , where associativity and monotonicity are somewhat modified. In order to accomplish this, we impose the following limitations on A and B: If , we assert the existence of a in A such that is in B and . The requirements on the proposed qualitative structure, which are summarized in the following definition, provide important integrity constraints for the production knowledge base.
Integrity constraints for qualitative structures. Let be a
nonempty set of qualitative descriptors (such as ``small'', ``medium'',
``large'') or , a
binary relation on , a nonempty subset of and
a binary function from into . The quadruple
is a qualitative structure if the following
six conditions are satisfied for all , , :