Kybernetes 23,3 10 General Structure Functions J. Montero, J. Tejada and J. Yáñez Universidad Complutense, Madrid, Spain Introduction Let us consider a system with n components. A central problem in reliability theory is how to determine the relationship between the performance level of the system itself and the performance levels of its components. This relationship is usually explained by a structure function. Much of the reliability literature develops the properties of binary systems with binary components, where only two states are supposed: perfect functioning “0” and complete failure “1’ (see, for example, Barlow and Proschan[1]). Hence, it is usually assumed that the system can be represented by a mapping φ: {0, 1} n →{0, 1}, called a structure function. The value φ(x 1 ,x 2 , …, x n ) is the state of the system when each component i is at level x i . But it is clear that such a binary assumption becomes a serious oversimplification when dealing with many real situations, since we find very often that components and systems can be in intermediate levels. Some authors have developed a theory of multistate systems in which a finite ordered set of performance levels is assumed. This restriction to a finite space of states also seems arbitrary in practice, since we can often discern a continuum of different states between both extreme performance levels. Recently, a number of articles (see, for example, Baxter[2,3] and Block and Savits[4]) have introduced another generalization by considering a class of structure functions in which the performance levels may take any value in a given real interval – for example the unit interval. These “continuum” structure functions are given by mappings φ: [0, 1] n →[0, 1]. In order to avoid confusion with topological continuity, these continuum structures have been named “fuzzy” structures in Montero[5] and Montero and Tejada[6]. In any case, it must be pointed out that at this moment we cannot properly talk about a unified theory for non-binary systems. A General Space of States for Structure Functions A general space of performance levels must allow us some kind of comparison between different states. Therefore, it can be assumed that at least a partially ordered set (L, ≥) has been associated to every space of states. That is, a pair given by a non-empty set L and a binary relation ≥ over L verifying reflexivity (a ≥ a 2200a ∈ L), anti-symmetry (a = b if a ≥ b and b ≥ a hold simultaneously) and transitivity (a ≥ c if a ≥ b and b ≥ c hold for some b ∈L ). In this way, when This research has been partially supported by Dirección General de Investigación Cientifica y Técnica, national grant number PB88-0137 and PB91 0389. Kybernetes, Vol. 23 No. 3, 1994, pp. 10-19. © MCB University Press, 0368-492X